FIXED POINT SETS AND TANGENT BUNDLES OF ACTIONS ON DISKS AND EUCLIDEAN SPACES by Bob Oliver The main result of this paper is the determination, for any given finite gro* *up G not of prime power order, of exactly which smooth manifolds can be fixed point sets* * of smooth G-actions on disks or on euclidean spaces. General techniques for constr* *ucting smooth actions on disks with fixed point set of a given homotopy type were deve* *loped in [O1], and the procedure for constructing actions on euclidean spaces is simi* *lar (but simpler). What is new here is a way of constructing a G-vector bundle over a G-* *complex of given homotopy type which extends a given G-bundle over the fixed point set.* * Such a G-bundle can then be used to control the process of "thickening up" the G-com* *plex to get a manifold with smooth G-action; and in particular to control the diffeomor* *phism type of the fixed point set. Here "G-complex" always means G-CW complex: a comp* *lex built up of orbits G=Hx Dn of cells (where G acts trivially on the disk Dn). The main technical result for constructing G-bundles, for a finite group G n* *ot of prime power order, is given in Theorem 2.4. Let P(G) denote the set of subgroup* *s of G of prime power order. Very roughly, given a finite G-complex X, a G-vector bu* *ndle j over XNP def=[H2=P(G)XH , and P -vector bundles P #X for all P 2 P(G), Theor* *em 2.4 gives conditions for being able to combine j and the P (after stabilization* *) to get a G-bundle over a G-complex X0 of the same (nonequivariant) homotopy type as X, and with (X0)NP = XNP . This result can then be combined with the equivariant thickening theorem of Edmonds & Lee [EL] and Pawalowski [Pa2] (see Theorem A.12 below), to construct manifolds with smooth G-action having given homotopy type * *and given tangential structure on the fixed point sets. Note that this procedure do* *es not (di- rectly) apply to construct closed manifolds with G-action, but only open (nonco* *mpact) manifolds, or compact manifolds with boundary. Instead of trying to formulate a general (and very messy) theorem about the * *con- struction of manifolds with smooth actions, we concentrate our applications her* *e to the case of smooth actions on disks and euclidean spaces. The study of this pr* *oblem goes back to P. A. Smith [Sm], who showed that the fixed point set of any conti* *nuous action of a p-group (for any prime p) on a finite dimensional Fp-acyclic space * *is itself Fp-acyclic. A converse to Smith's theorem was proven by Lowell Jones [Jo], who * *showed among other things that any compact smooth stably complex Fp-acyclic manifold c* *an be the fixed point set of a smooth action of the group of order p on a disk. T* *hus, if G is any nontrivial p-group, then a compact smooth manifold can be the fixed po* *int set of a smooth G-action on a disk if and only if it is stably complex and Fp-a* *cyclic. A similar (but simpler) construction can be used to prove the corresponding res* *ult for smooth p-group actions on euclidean spaces. Typeset by AM S-* *TEX 1 Later examples by various authors (cf. [Br, xI.8]) showed that when G does n* *ot have prime power order, then the situation is much less rigid. For example, one can* * find a smooth action of any such G on some euclidean space with fixed point set havi* *ng the homotopy type of any given countable finite dimensional complex. However, * *the situation for actions on disks is more complicated. The main result in [O1] sa* *ys that there is an integer nG 0 with the property that a finite CW complex F is homot* *opy equivalent to the fixed point set of some smooth G-action on a disk if and only* * if O(F ) 1 (mod nG ). The results here now make it possible to determine exactly* * which manifolds can be fixed point sets of smooth actions on disks or on euclidean sp* *aces; and also to describe (at least stably) the possibilities for the normal bundle * *of the fixed point set. These results are summarized in the following theorem: Theorem 0.1. Let G be any finite group not of prime power order. Fix a smooth manifold F and a G-vector bundle j#F satisfying the following three conditions: (1) j is nonequivariantly a product bundle; fi (2) for each prime pfi|G| and each p-subgroup P G, [j|P ] is infinitely p-divi* *sible in gKOP (F )(p)(where gKO P(F ) = KOP (F )=KOP (pt)); and (3) jG ~= o(F ) (the tangent bundle of F ). Then there is a smooth action of G on a contractible manifold M such that MG =* * F , and such that o(M)|F ~= j (V xF ) for some G-representation V with V G = 0. * *If @F = ;, then M can be chosen to be a euclidean space. If F is compact and O(F )* * 1 (mod nG ), then M can be chosen to be a disk. Conditions (1)-(3) in Theorem 0.1 are also necessary: if G acts smoothly on* * any contractible manifold M, then they hold for the pair (F; j) = (MG ; o(M)|MG ). * *Note in particular point (2): [j|P ] is infinitely p-divisible in gKOP (F ) since j|P i* *s the restriction of a P -bundle over the Fp-acyclic manifold MP (hence the group gKOP (MP ) is u* *niquely p-divisible). Theorem 0.1 still leaves it rather unclear exactly which manifolds can be th* *e fixed point set of a G-action on a disk or euclidean space. In order to make this mor* *e precise, we first need some definitions. Let MC MC+ MR be the classes of finite groups for which there exist G-representations V and W which are complex, self-conjuga* *te, or real, respectively, such that V |P ~=W |P for any P G of prime power order, an* *d such that dim (V G) = 1 and dim (W G) = 0. By Lemma 3.1 below, G 2 MC if and only if it contains an element not of prime power order, G 2 MC+ if and only if it con* *tains an element not of prime power order which is conjugate to its inverse, and G 2 * *MR if and only if it contains a subquotient which is dihedral of order 2n for some* * n not a prime power. (This condition for a group to be in MR was pointed out to me by E* *rkki Laitinen.) In the following theorem, for any abelian group A, we let qdiv(A) (the subgr* *oup of "quasidivisible" elements) denote the intersection of all kernels of homomor* *phisms from A into free abelian groups. When A is finitely generated, this is just the* * torsion subgroup of A. Also, the standard induction and forgetful maps between the grou* *ps of real, complex, and quaternion vector bundles over a space X are denoted as foll* *ows: 0 gKO (X) ________wcu________eK(X)]________u________cKSp(X): r q w 2 Theorem 0.2. Let G be a finite group not of prime power order. Let Fix(G) be the class of smooth manifolds F (without action) for which there is a G-bundle * *j#F satisfying conditions (1)-(3) in Theorem 0.1. Then a smooth manifold F is the * *fixed point set of a smooth action of G on a euclidean space if and only if F 2 Fix(G* *) and @F = ;; while F is the fixed point set of a smooth action of G on a disk if and* * only if F is compact, F 2 Fix(G), and O(F ) 1 (mod nG ). Furthermore, Fix(G) is descr* *ibed as follows (where Syl2(G) is a Sylow 2-subgroup of G): F 2Fix(G) () || Syl2(G) 6C G | Syl2(G) C G * * || ______________||_____________________________________|________________________* *_||__ G2 MR || (A) (no restriction) | ___ || ______________||_____________________________________|______________________||* *__(B) G2 MC+ rMR || c([o(F )]) 2 c0(]KSp (F ))+ qdiv(Ke(F|)) ___ || ______________||_____________________________________|______________________||* *__(C)(D) || e g | [o(F )] 2 r(Ke(F )* *)|| G2 MCr MC+ |||| [o(F )] 2 r(K (F ))+ qdiv(KO (F |))| (F is stably complex* *)|||| ______________||_____________________________________|________________________* *_||_(E)(F) G62MC || [o(F )] 2 qdiv(KgO (F )) | [o(F )] 2 r qdiv(Ke(F* * )) || ______________||_____________________________________|________________________* *___ || This theorem extends results of Edmonds & Lee [EL, Theorem A] and Pawalowski [Pa2, Theorems 5.6 & 5.9]. However, all of their constructions give fixed point* * sets with stably complex tangent bundle, and the possibility of having other fixed point * *sets (when Syl2(G) 6C G) is new. Note that if G =2MC, then all connected components of the* * fixed point set of a G-action on a disk or euclidean space must have the same dimensi* *on; while if G 2 MCr MR then the dimensions of the components can be different but must have the same parity (see [Pa1, Theorem A]). In contrast, if G 2 MR, then * *the components of the fixed point set can have arbitrary dimensions. Examples of groups in the above classes include: (A) D(2n), (B) Q(4pa), (* *C) D(2pa)x Cqb, (D) Cn, (E) D(2pa), (F) F2io Cpa where pa|2i- 1. Here, in all c* *ases, n denotes any integer not a prime power, p and q denote distinct odd primes; an* *d Cm , D(m), and Q(m) denote cyclic, dihedral, and quaternion groups of order m. The conditions on F in cases (B) and (C) above are very similar, and it is * *not immediately clear that they give distinct classes Fix(G). To see that they do,* * set X = S5[ j2e8: the complex obtained by attaching an 8-cell to S5 via the nontri* *vial element j2 2 ss7(S5). We leave it as an exercise to check that gKO (X) ~=Z, and* * that the maps KgO (X) - --c-! eK(X) - -c0-- ]KSp(X) ~= ~= are isomorphisms. Thus, if F is a compact manifold with the homotopy type of X * *such that [o(F )] generates gKO(F ), then F 2 Fix(G) for G of type (B), but not for * *G of type (C). So far, we have only discussed the case of actions of finite groups. If G is* * a compact Lie group with identity component G0 which acts smoothly on a contractible mani* *fold M with fixed point set F , then one can show (using [JO, Proposition 4.6] and t* *he definition of Fix(-)) that F 2 Fix(G=G0). More precisely, (o(M)|F )G0 is a G=* *G0- bundle which satisfies conditions (1)-(3) in Theorem 0.1. In particular, if G i* *s connected 3 and nonabelian, then F can be the fixed point set of a smooth action of G on a * *disk if and only if it is stably parallelizable (see [O2, Theorems 3 & 5]). This is* *, however, still far from answering the question of which manifolds can be fixed point set* *s, since in general more homotopy types can occur as fixed point sets of G-actions on disks* * than of G=G0-actions on disks. Since the numbers nG play such a key role in the above theorem, we summarize* * here their computation in [O1, Corollary to Theorem 5] and [O3, Theorem 7]. Let G1 * *be the class of all finite groups G which contain a normal subgroup P C G of prime* * power order such that G=P is cyclic. For each prime p, let Gp denote the class of al* *l finite groups G which contain a normal subgroup in G1 of p-power index. Theorem 0.3. Fix a finite group G not of prime power order. For any prime p, p|* *nG if and only if G 2 Gp. Thus, nG = 0 if and only if G 2 G1, and nG = 1 if and o* *nly if G =2[ pGp. In general, nG is equal to 0, 1, a product of distinct primes, * *or 4; and nG = 4 if and only if (1) G lies in an extension 1 -! Cm -! G -! C2k -! 1, where Cm is cyclic of od* *d order m and C2k is cyclic of order 2k, (2) G =2G1, but its subgroup of index 2 does lie in G1, and (3) there is no unit u 2 (Zim )* such that ff(u) = -u, where im is a primitive * *m-th root of unity, and ff 2 Gal(Qim =Q) is induced by the conjugation action of a gen* *erator of G=Cm ~= C2k. As another special case of Theorem 0.1, we note the the following theorem ab* *out tangential representations at isolated fixed points. This generalizes results o* *f Edmonds & Lee [EL, Theorem B] and Pawalowski [Pa1, Theorem B]: Theorem 0.4. Let G be any group not of prime power order. Let V0; V1; : :;:Vm * * be (real) G-representations such that V0|P ~=V1|P ~=. . .~=Vm |P for any P G of p* *rime power order, and such that ViG = 0 for all i. Then there exists a G-representat* *ion W with W G = 0, and a smooth action of G on a euclidean space (or a disk if nG |m* *) with exactly m+ 1 fixed points x0; : :;:xm , such that the tangential representation* * at xi is Vi W . The paper is organized as follows. In Section 1, a space B*GO is constructed* * which has the following property (Proposition 1.3): for any finite G-complex X, [X; B* **GO]G is (roughly) the inverseflimitiof the groups KOP (X)(p), taken over all p-subgr* *oups P G and all primes pfi|G|. The problem of lifting maps X -! B*GO to BG O is th* *en handled in Section 2, and this leads to a general criterion (Theorem 2.4) for c* *onstructing G-bundles over G-complexes of given (nonequivariant) homotopy type and with giv* *en fixed point data. The proofs of Theorems 0.1 and 0.2, and some examples, are t* *hen given in Section 3. Finally, in an appendix, some technical results are listed* *, most of which are well known but seem hard to find in the literature. The equivariant t* *hickening theorem in the version of Pawalowski is also stated there (Theorem A.12). The proof of these results _ more precisely the constructions in Section 1 _* * were to a great extent motivated by my joint work with Stefan Jackowski on vector bundl* *es over classifying spaces of compact Lie groups [JO]. The connections with [JO] have l* *argely 4 disappeared while this work has evolved, but it probably would not have been po* *ssible without the discussions we had while writing that paper. 1. An approximation to the classifying spaces for G-bundles Throughout this section, G will be a fixed finite group. Let O(G) denote the* * orbit category of G: the category whose objects are the orbits G=H for all subgroups * *H G, and where Mor O(G)(G=H; G=K) is the set of all G-maps G=H -! G=K. For each prime p, Op(G) O(G) will denote the full subcategory whose objects are the orbits G=* *P for p-subgroups P G. Also, O1(G) denotes the full subcategory with one object G=1. For any full subcategory C O(G), we define EC = hocolim-----!G=H : G=H2C This can be regarded as the nerve of the category whose objects are the cosets * *aH for G=H 2 C, and where there is one morphism from aH to bK for each C-morphism G=H -! G=K which sends aH to bK. (In particular, there is at most one morphism between any pair of objects.) From this definition, EC is seen to be a G-comple* *x all of whose orbit types lie in C. Also, ECH is contractible for any G=H in C, since * *it is the nerve of a category with initial object the coset eH. In particular, EO1(G) ~=E* *G. More generally, by equivariant obstruction theory, EC is "universal" among G-complex* *es with orbits in C: for any such X there is a G-map X -! EC which is unique up to G-ho* *motopy. In the appendix, BG O is defined to be the infinite mapping cylinder of ma* *ps BG O(0) -! BG O(d) -! BG O(2d) -! : :,:where d = |G|, where BG O(n) is the base space of the universal n-dimensional G-bundle, and where the maps are sta- bilization by the regular representation RG. Bundle direct sum define* *s product maps BG O(n)x BG O(m) -! BG O(n+ m), and these combine to define a G-* *map BG Ox BG O -! BG O which makes BG O into a G-equivariant H-space.i Alternativel* *y,j ` 1 one can define BG O as the identity component of the loop space B n=0BG O(n) (once the BG O(n) have been defined precisely enough to make their disjoint uni* *on into a topological monoid); and then the H-space structure on BG O is automatic. * * fi We will also have need for the p-localization BG O(p)of BG O, for any prime * *pfi|G|. One elementary way to define this is as the infinite mapping cylinder of the ma* *ps BG O -.n1-!BG O -.n2-!BG O -.n3-!BG O -.n4-!:;: : where BG O -.n!BG O is multiplication by n (using the H-space structure), and w* *here n1; n2; : :i:s any sequence of positive integers prime to p such that each prim* *e dif- ferent from p divides infinitely many of the ni. Thus, for any finite G-comple* *x X, [X; Z(p)xBG O(p)]G ~=KOG (X)(p). We regard BG O as a subcomplex of BG O(p)via inclusion into the first term of the cylinder. By construction, for each subgro* *up H G, (BG O(p))H is the p-localization of (BG O)H (where the group of components ha* *s also been p-localized). Hence by equivariant obstruction theory, it is immediate th* *at the equivariant H-space structure on BG O extends to an equivariant H-space structu* *re on BG O(p). 5 Definition 1.1. Define the G-space B*GO to be the pullback in the diagram: Y B*GO _______________w map (EOp(G); BG O(p)) | | p||G| | | | | | | | |u Y |u map (EO1(G); BG O) ________wdiagmap(EO1(G); BG O(p)); p||G| where the right hand vertical map is induced by restriction to EO1(G) (regarded* * as a subspace of EOp(G)). Let LG : BG O ,--- -! B*GO be the G-equivariant map induced by inclusions into constant maps in the above * *square. By the homotopy extension property for inclusions of simplicial complexes, t* *he right hand vertical map in the above square satisfies the equivariant homotopy liftin* *g prop- erty. Thus, B*GO is also a homotopy pullback of that square. We want to study maps from finite G-complexes to B*GO. The following lemma, a special case of a theorem of [JM], will be needed to handle the higher inverse * *limits which arise as obstructions. fi Lemma 1.2. For any finite G-complex X and any prime pfi|G|, i j lim-j KOG (G=P xX)(p) = 0 for all j > 0. G=P2Op(G) Proof. A contravariant functor F : Op(G) -! Ab is called a Mackey functor if t* *here is a covariant functor F* : Op(G) -! Ab which takes the same values on objects, an* *d such that any pullback square ak G=Ki ________wff1G=H1 i=1 | | | |ff2 |fi1 | | |u |u G=H2 _________wfi2G=H induces a commutative square Mk F (G=Ki) ________wF*(ff1)F (G=H1)u i=1 u| | | | |F(ff2) |F(fi1) | | | | F (G=H2) _________wF*(fi2)F (G=H): 6 By a theorem of Jackowski and McClure [JM, Proposition 5.14], for any Mackey fu* *nctor F : Op(G) -! Z(p)-mod , lim-j(F ) = 0 for all j > 0. Now let F be the contravariant functor F (G=H) = KOG (G=P xX)(p). Then any map f : G=H -! G=H0 in O(G) induces a homomorphism F* : F (G=H) -! F (G=H0): defined by sending a G-bundle #(G=Hx X) toLthe G-bundle 0#(G=H0x X) such that the fiber over any (a0; x) 2 G=H0x X is a2f-1a0(a;x). This makes F into a Mac* *key functor; and hence its higher limits over Op(G) vanish by [JM]. * *____ It will be convenient, when X is a (finite) G-complex, to write * *KO G (X) = [X; BG O]G ; i.e., the group of virtual G-bundles which have virtual dimension * *zero over all connected components of X. We are now ready to prove the following proposit* *ion, which describes how to construct maps from a finite G-complex X to B*GO. Proposition 1.3. For any finite G-complex X, the square Y ____ [X; B*GO]G ___________w lim KO P(X)(p) | G=P-2Op(G)| | p||G| | | | | | (* *1) | | ____|u Y __|u_ KO (X)G ___________________w KO (X)(p) G p||G| is_a_pullback_square._ Here, functoriality on the right is induced by the ident* *ification KO P(X) ~=KO G(G=P xX); and the right hand vertical arrow is induced by restr* *icting the limit to the subcategory O1(G) Op(G) and identifying lim-O1(G)(-) with (-)* *G . Proof. The basic idea of the proof is to regard map G(X; B*GO) as the homotopy * *inverse limit, over an appropriate category, of the spaces map P (X; BG O(p)) (for p-su* *bgroups P G) and map (X; BG O). We want to show that [X; B*GO]G is the inverse limit o* *f the corresponding sets of components; and this follows upon showing that certain hi* *gher inverse limits vanish. The argument given here is a more direct version of this* * idea; and is similar to the approach used by Wojtkowiak [Wo] to describe maps from a homo* *topy direct limit to a space. Square (1) above is equivalent to the diagram Y [X; B*GO]G ______________wXS(X) _______________w lim [-x X; BG O(p)]G | O-p(G) | | p||G| | | | | | | | (* *2) |u Y |u lim-[Gx X; BG O]G ________w lim-[Gx X; BG O(p)]G ; O1(G) p||G|O1(G) where S(X) is defined to be the pullback. We must show that X is a bijection.* * In Step 1, certain cochain complexes D*(X; n) are defined, and their homology grou* *ps are 7 shown to vanish. And in Step 2, the obstructions to constructing maps X -! B*GO* * (or to constructing a homotopy between two such maps) are shown to be homology grou* *ps of the D*(X; n). Step 1: For any category C and any contravariant functor F : C -! Ab , let C*(* *C; F ) denote the cochain complex i Y Y Y j C*(C; F ) = 0 -! F (c) -! F (c0) -! F (c0) -! : :;: c c0-!c1 c0-!c1-!c2 where the differentials are alternating sums of face maps. The homology groups* * of C*(C; F ) are the higher limits lim-*(F ) (cf. [O4, Lemma 2]). For each n 1, FnX : O(G) -! Ab will denote the functor h restr. i FnX(G=H) = Coker KOG (G=Hx Dn+1 xX) ---! KOG (G=Hx Sn xX) ~=KgO H(n(X+ )): Let D*(X; n) be the cochain complex defined via the short exact sequence Y 0 -! D*(X; n) --- -! C*(O1(G); FnX) C* Op(G); FnX(-)(p) p||G| Y (* *3) - --- ! C* O1(G); FnX(-)(p) -!0: p||G| For all j > 0, lim-jOp(G)FnX(-)(p) = 0 by Lemma 1.2 (applied to the G-complexes Sn xX and X). Also, a functor M : O1(G) -!QAb is the same as a Z[G]-module, and lim-*O1(G)M ~= H*(G; M). Since Hj(G; M) ~= p||G|Hj(G; M(p)) for any Z[G]-module M, the long exact cohomology sequence for (3) reduces to an exact sequence n G Y 0 n 0 -! H0 D(X; n) --! gKO( (X+ )) lim-gKO P( (X+ ))(p) p||G|Op(G) (* *4) --'-! Y KgO (n(X G 1 + ))(p) --! H D(X; n) -! 0; p||G| and Hj D(X; n) = 0 for all j 2. For any P G, the composite gKO (n(X+ )) -transfer----!gKOP(n(X+ )) -restr.--!gKO(n(X+ )) is the norm homomorphism for the action of P , and in particular sends any x 2 KgO (n(X+ ))G to |P |.x. Here, the transfer map sends a vector bundle over n(X+* * ) to the direct sum of its translates under the action of P (considered as a P -bund* *le). Thus, if pm is the highest power of p dividing |G|, then h i Im lim-0gKOP(n(X+ )) --- ! KgO (n(X+ ))G pm .gKO(n(X+ ))G : Op(G) 8 Since gKO(n(X+ ))G maps onto the sum of the Z=pm KgO (n(X+ ))G , this shows th* *at the map ' in (4) is surjective, and hence that H1 D(X; n) = 0. Step 2: We now consider maps from X to B*GO. For each 0 n 1, let Un(X) be the space defined by the pullback square Y Un(X) _________________w map (EOp(G)(n)xX; BG O(p)) | | | p||G| | | | | | | | (* *5) |u Y |u map (EO1(G)(n)xX; BG O) ________wdiagmap (EO1(G)(n)xX; BG O(p)) : p||G| Here, EC(n) (for C = Op(G) or O1(G)) denotes the n-skeleton of the complex a i a j OE EC = (G=H0x n) ~; (* *6) n0 G=H0-!:::-!G=Hn with the usual identifications induced by face and degeneracy maps. By Definiti* *on 1.1, U1 (X) ' map (X; B*GO). By the pullback square in (2), an element of S(X) corresponds to a choice of* * G- maps Gx X -! BG O, and G=P xX -! BG O(p)for all p and all p-subgroups P G: maps which agree up to homotopy with respect to morphisms in Op(G). In other words, by (6), we can identify S(X) with Im [ss0(U1(X)) -! ss0(U0(X))]. To show* * that X : [X; B*GO]G -! S(X) is onto, we must thus show that any element of U1(X) can be lifted to an element of U1 (X) which has the same image in ss0(U0(X)). Fix an element f1 2 U1(X), and consider the obstructions to lifti* *ng it to U2(X). By (6) again, a 2-simplex in EOp(G) corresponds to a sequence of maps G=P0-! G=P1-! G=P2 in Op(G), and the obstruction to extending f1 to that 2-simp* *lex lies in the group h____ ____ i Coker KO G (G=P0x D2x X)(p)-- ! KO G(G=P0x S1x X)(p) = F1X(G=P0)(p): Similarly, the obstruction to extending f1 to any 2-simplex in EO1(G) lies in F* *1X(G=1). These individual obstructions combine to give an element ff2 2 D2(X; 1) as the * *total obstruction to lifting f1 to some f2 2 U2(X). This element is easily seen to be* * a cocycle, and hence is a coboundary by Step 1. And if ff1 = ffi(fi1) for fi1 2 D1(X; 1),* * then f1 can be changed on 1-simplices (in a way specified by fi1) to remove the obstruc* *tion; after which the "modified" map can be lifted to an element f2 2 U2(X). (Note t* *hat the co-H-space structure on the suspensions induces the usual addition on the g* *roups KgO P((X+ )).) Upon continuing this process, we see that at each stage the obst* *ruction to lifting fn 2 Un(X) to fn+1 2 Un+1 (X) (while allowing fn to be changed on n- simplices) lies in Hn+1 (D*(X; n)), which again vanishes by Step 1. This shows that X : [X; B*GO]G -! S(X) is onto. To show that it is injectiv* *e, we start with two elements f; f0 2 map G(X; B*GO) ~=U1 (X), together with a homoto* *py 9 F0 2 U0(Xx I), and then lift the homotopy one step at a time. For each n 0, t* *he obstruction to lifting a homotopy Fn 2 Un(Xx I) to Un+1 (Xx I) (while taking a * *given value on Xx {0; 1}) lies in Hn+1 (D*(X; n+1)). And this again vanishes by Step * *1. Proposition 1.3 does not in general hold for infinite G-complexes. But it do* *es hold for countable complexes with fixed G-action. The following corollary to Proposition 1.3 will be needed in Section 3. Corollary 1.4. B*GO can be given the structure of a G-equivariant H-space in a * *way such that LG : BG O -! B*GO is an H-space homomorphism. Also, for any finite * *G- complex X, [X; B*GO]G is an abelian group with the property that (Z=n) [X; B*G* *O]G is finite for all n > 0. Proof. The H-space structure on B*GO, and LG being an H-space homomorphism, follow immediately from the pullback square in Definition 1.1, together with th* *e H- space structures on BG O and its localizations (see the discussion before Defin* *ition 1.1). By Proposition 1.3, for any finite G-complex X, there is an exact sequence ____ Y i ____ j Y ____ G 0 -! [X; B*GO]G -! KO (X)G lim-KO P(X)(p) -! KO (X)(p) : (* *1) p||G|Op(G) p||G| So [X; B*GO]G is abelian, and (Z=n) [X; B*GO]G is finite for any n > 0 since (* *Z=n) - and Tor(Z=n; -) are finite for the other two terms in (1). 2. Construction of G-bundles Proposition 1.3 describes a procedure for constructing G-maps from a finite * *G- complex X to B*GO. What we really are interested in is the construction of G-ma* *ps from X to BG O. In general, of course, G-maps from B*GO cannot be lifted to BG O (LG* * is not a G-homotopy equivalence). To get around this, we prove a rather complicated li* *fting result (Proposition 2.3); and then apply it in Theorem 2.4 to construct G-vecto* *r bundles by "pasting together" bundles over certain subcomplexes and for certain subgrou* *ps. The first step is to compare the homotopy groups of fixed point sets in B*GO* * with those in BG O. Let RO (G) ~=KOG (pt) denote_the orthogonal representation ring * *of G, and let IRO (G) = Ker[RO (G) -dim-!Z] ~=KO G(pt) be its augmentation ideal. Lemma 2.1. Let G be any finite group, and let LG : BG O -! B*GO be the map of Definition 1.1. Then the following hold. (a) LG is (nonequivariantly) a homotopy equivalence. fi (b) Fix a prime pfi|G| and a p-subgroup 1 6= P G. Then ss0((B*GO)P ) ~=IRO * *(P )(p), and ss0(LPG) is isomorphic to the inclusion of IRO (P ) into IRO (P )(p). For * *each x 2 (BG O)P , ssi(LPG; x) : ssi (BG O)P ; x ----! ssi (B*GO)P ; LG (x) is an isomorphism for i = 1; and for i > 1 its kernel and cokernel are torsion * *prime to p with the additional property that their m-torsion subgroups are finite for al* *l m. 10 fi Proof. Fix a prime pfi|G| and a p-subgroup P G. By Proposition 1.3 (applied wi* *th X = G=P ) there is a pullback square Y ____ [G=P; B*GO]G ~=ss0((B*GO)P ) _________w lim KO Q (G=P )(q) | G=Q-2Oq(G)| | q||G| | | | | | (* *1) | | ____ |u Y ____ |u KO (G=P )G _______________w KO (G=P )(q)G : q||G| Similarly, for any i > 0, Proposition 1.3 yields a pullback square [i(G=P+_);_B*GO]G_ ss~((B* O)P ) __________wY lim KO-i(G=P ) [pt; B*GO]G = i G| G=Q2-Oq(G)| Q (q) | q||G| | | | | | (* *2) | | |u Y |u KO-i(G=P )G ______________w KO-i(G=P )(q)G : q||G| For any i 0 and any Q G, KO-iQ(G=P ) ~=KO-iG(G=P x G=Q) ~=KO-iP(G=Q): In general, if FP is any contravariant functor from P -complexes to abelian gro* *ups such that FP (Xq Y ) = FP (X) FP (Y ), then aeF (pt) if q = p lim- FP (G=Q) ~= lim-(FP ) ~= P P (* *3) G=Q2Oq(G) Oq(P) FP (P ) if q 6= p. The first isomorphism holds since for each P -orbit P .gQ in any G=Q, there is * *an Oq(G)- morphism G=Q0 -! G=Q (where Q0 = P \gQg-1 ) which sends the P -orbit P=Q0 in Oq(P ) isomorphically to P .gQ. Also, Oq(P ) has a final object if p = q, and * *contains only the free orbit if p 6= q. ____ If we now apply (3) with FP = KO P (-)(q), then square (1) is reduced to an * *isomor- phism ~ ____ ss0 (B*GO)P -- =--! KO P(pt)(p)~=IRO (P )(p) ____ (note that KO (G=P ) = [G=P; BO] = 0). Thus, ss0(LPG) is isomorphic to the inc* *lusion of IRO (P ) into IRO (P )(p). And if (3) is applied with FP = KO-iP(-)(q), then* * square (2) reduces to a pullback square ssi((B*GO)P ) ___________wKO-iP(pt)(p) | | | | (* *4) |u |u KO-i(pt) ____________wKO-i(pt)(p): 11 When P = 1, this shows that ssi(B*GO) ~=KO-i(pt) ~=ssi(BG O) for all i > 0, and* * hence that LG is nonequivariantly a homotopy equivalence. -i forget Now set Mi = Ker KOP (pt) --- --i KO-i(pt) . The forgetful map is a surjec- tion: split by regarding a bundle over Si as a P -bundle with trivial action. S* *o by (4), the kernel and cokernel of the homomorphism P ssi(LPG) * P KO-iP(pt) ~=ssi (BG O) -- --- ! ssi (BG O) are given by P Ker(ssi(LPG)) ~=Ker Mi-!Mi (p) and Coker (ssi(LG )) ~=Coker Mi-!Mi (p): (* *5) In particular, since Miis finitely generated, Ker ssi(LPG) and Coker ssi(LPG) * * are torsion of order prime to p, and have finite m-torsion for any m > 0. It remains to show that ss1(LPG) is an isomorphism. By Propositio* *n A.2(b), KO-1P(pt) ~= ss1((BG O)P ) is a sum of one copy of Z=2 for each irr* *educible P - representation of real type. So if p = 2, then ss1(LPG) is an isomorphism by (5* *), since M1 is a finite 2-group. If p is odd, then the only irreducible P -representation o* *f real type is the trivial one (see Proposition A.1(c)); so KO-iP(pt) ~=KO-i(pt) ~=Z=2, M1 = 0* *, and ss1(LPG) is again an isomorphism. When f : X -! Y is a given map between spaces, we will frequently write ssi(* *Y; X; x) (for x 2 X) to denote the relative homotopy group ssi(Zf; X; x). Here, Zf is th* *e mapping cylinder of f. Also, we write ssi(Y; X) when X is connected and the basepoint n* *eed not be specified. The next lemma will provide the induction step in our construction of G-bund* *les. Lemma 2.2. Fix a finite group G and a prime p. Let X -ff!Z -fi!Y be G-maps, whe* *re (1) X and Y are countable finite dimensional (nonempty) G-complexes; (2) Z and Y are connected, ss1(Y; Z) = 1, ss2(Y; Z) is abelian, and ssi(Y; Z) * * Z(p)= 0 for all i 2; and (3) for any nontrivial p-subgroup 1 6= P G, (fiff)P : XP -! Y P is an Fp-hom* *ology equivalence. __ Then there exist a countable finite dimensional G-complex X X and an extension _ff: __X-! Z of ff, such that G acts freely on __XrX and such that fif_f: __X-!* * Y is an Fp-homology equivalence. If in addition, (4) X and Y are finite G-complexes, (5) Ker(ss1(fi)) has finite n-torsion for all n, and (6) O(XH ) = O(Y H) for all cyclic subgroups H G of order prime to p, __ then X can be chosen to be a finite G-complex. Proof. Finite case: Assume that all points (1)-(6) hold. By attaching some f* *inite number of free orbits of 1-cells to X if necessary (and extending ff accordingl* *y), we can 12 assume that X is connected and that ss1(ff) : ss1(X) i ss1(Y ) is onto. Set ss1(fiff) ss1(fi) KX = Ker ss1(X) --- --i ss1(Y ) and KZ = Ker ss1(Z) --- -i ss1(Y )* * : Since ss1(X) and ss1(Y ) are both finitely presented, KX is finitely generated * *as a normal subgroup of ss1(X) (cf. [Ro, Lemma 1.43(i)]). By (2), KZ is abelian and torsion* * prime to p, and by (5) its n-torsion subgroup is finite for any n. Then Im[KX -ff*!KZ ] * *is finite (it has bounded torsion since KX is finitely generated); and hence Im[ss1(X) -ss1ff* *-!ss1(Z)] is finitely presented (an extension of one finitely presented group by another). S* *o by [Ro] again, Ker (ss1(ff)) is finitely generated as a normal subgroup of ss1(X). We * *can thus attach finitely many free orbits of 2-cells to X, to obtain a finite G-complex * *X1 X and a map ff1 : X1 -! Z extending ff, such that ss1(ff1) is injective, and Ker(* *ss1(fiff1)) is finite abelian of order prime to p. Set d = max {dim (X); dim(Y ); 2}. We next construct a sequence of finite co* *mplexes X1 X2 X3 . . .Xd; together with G-maps ffm : Xm -! Z extending ff1, such that each Xm (for 2 m d) is constructed from Xm-1 by attaching free orbits of m-cells, and such t* *hat ssi(Ze; eXm) Z(p)= 0 for each i m. Assume that Xm-1 has been constructed. Th* *en by Lemma A.7, ssm (Ze; eXm-1) Z(p)~=ssm (Ye; eXm-1) Z(p) is finitely generated as a Z(p)[ss1(Xm-1 )]-module. We can thus attach some fin* *ite number of free orbits of m-cells to Xm-1 (while extending ffm-1 ) to get a new comple* *x Xm such that ssm (Ze; eXm) Z(p)~=ssm (Ye; eXm) Z(p)= 0. Now consider the sequence of maps Xd -ffd! Z -fi! Y . By assumpt* *ion, ssi(Y; Xd) Z(p) = 0 for all i d = dim (Xd) dim (Y ). Also, by Lemma A.7, ap- plied to the pairs (Y; Xd) and (Y; Z), Hi(Y; Xd; Z(p)) ~= Hi(Z; Xd; Z(p)) = 0 f* *or all i d, and the Hurewicz homomorphism ssd+1(Z; Xd) Z(p)~=ssd+1(Y; Xd) Z(p)-- --- i Hd+1(Y; Xd; Z(p)) is surjective. By Lemma A.10, together with conditions (3) and (6), Hd+1(Y; Xd;* * Fp) is free as an Fp[G]-module. We can thus choose elements of ssd+1(Z; Xd) which repr* *esent a basis of Hd+1(Y; Xd; Fp), and attach accordingly_free orbits of_(d + 1)-cells* * to Xd, extending ffd, to get a finite G-complex X Xd and a map _ff: X -! Z which is* * an Fp-homology equivalence. Countable case: The proof is similar in the countable case (i.e., when we on* *ly assume conditions (1)-(3)), but much simpler. All homotopy groups of Y and the* * Xi are countable by Lemma A.6. So at each stage, the relevant homotopy elements ca* *n be eliminated by attaching only countably many cells. And in the last step, Hd+1(Y* *; Xd; Fp) is stably free as a countably generated projective Fp[G]-module by Lemma A.10, * *and so free orbits of d- and_(d+_1)-dimensional cells can be attached to Xd to cons* *truct the Fp-homology equivalence X -! Z. We are now ready lift maps from B*GO to BG O. 13 Proposition 2.3. Assume that G is a finite group not of prime power order. Let X ________wfBG O | ' | | | LG| (* *1) |u |u Y ________wfYB*GO be a homotopy commutative square of G-maps, wherefXiand Y are countable finite dimensional G-complexes. Assume, for each prime pfi|G| and each p-subgroup 1 6=* * P G, that P (fPY)* * P P (LPG)* * P Im ss0(Y ) ---! ss0((BG O) ) Im ss0((BG O) ) ----! ss0((BG O) ) :* *(2) __ Then there is_a_countable finite dimensional G-complex X X such_that all isot* *ropy subgroups_of_X r X have prime power order, and extensions _' : X -! Y of ' a* *nd f : X -! BG_O of f such that _' is (nonequivariantly) a homotopy equivalence* * and fY O_'' LG Of. If, furthermore, (3) X and Y are finite G-complexes with XG 6= ; and Y connected, (4) O(XH ) = O(Y H) for all H G not of prime power order, and fi (5) O ('-1 YiP)H = O (YiP)H for each prime pfi|G|, each p-subgroup 1 6= P G,* * each connected component YiP of Y P, and each cyclic subgroup 1 6= H=P N(P )=P of order prime to p, __ then X can be chosen to be a finite G-complex. Proof. We concentrate on the proof in the case where X and Y are finite complex* *es and conditions (3)-(5) hold. The proof in the finite dimensional case is simpler, a* *nd some remarks will be made afterwards as to how it differs from that for finite compl* *exes. Define Z to be the (G-equivariant) homotopy pullback in the following square: Z ________wflBG O | f|i | | LG| (* *6) |u |u Y ________wfYB*GO: Lemma 2.1(b) and Condition (2) imply that for any P G of prime power order, P ) ss1(fiP ) ss0(ZP ) -ss0(fi---!~ss0(Y P) and ss1(ZP ; x) --- -i ss1(Y P; fi(x))(* *8x72)ZP : = onto By the homotopy commutativity of (1), there is a G-map ff : X -! Z such that fi* * Off ' ' and fl O ff ' f. Finite case: Step 1: Let P1; : :;:Pk be conjugacy class representatives f* *or all subgroups 1 6= P G of prime power order, ordered from largest to smallest (i.* *e., i j if Pi contains a subgroup conjugate to Pj). We first construct finite G-co* *mplexes 14 X = X0 X1 . . .Xk, together with maps ffi : Xi -!Z (where ff0 = ff), satisfyi* *ng the following conditions for all 1 i k: (a) Xir Xi-1 contains only orbits of type G=Pi, (b) ffi|Xi-1 = ffi-1, and (c) (fiOffi)Pi : (Xi)Pi -! Y Piis an Fpi-homology equivalence, where pi is t* *he prime such that Pi is a pi-group. Fix some 1 i k, and assume that Xi-1 and ffi-1 have been constructed. Cons* *ider the maps Pi fiPi (Xi-1)Pi (ffi-1)-----!ZPi ----! Y Pi: After restricting to any connected component of Y Pi(and to those connected com* *po- nents of ZPi and (Xi-1)Pi which map into it), these maps satisfy hypotheses (2)* *-(6) of Lemma 2.2, applied with the action of the group N(Pi)=Pi. Note in particular* * that conditions (2) and (5) in Lemma 2.2 follow from (7) and Lemma 2.1 (and the pull* *back square (6)), that condition (3) follows from the assumptions on Xi-1, and that * *condition (6) follows from condition (5) here. So by Lemma 2.2, there is a finite N(Pi)=P* *i-complex W (Xi-1)Pi, and an equivariant map bff: W -! ZPi which extends (ffi-1)Pi, such* * that fiPiObffis an Fpi-homology equivalence. And if we set Xi = Xi-1[ Gx (incl)(Gx N* *(Pi)W ) and ffi = ffi-1[ (Gx bff), then the pair (Xi; ffi) satisfy conditions (a), (b),* * and (c) above. Step 2: It remains to deal with the free orbits. Note that fi : Z -! Y is (* *nonequiv- ariantly) a homotopy equivalence since LG is (by Lemma 2.1). Since Xk and Y are* * both finite, we can attach free orbits of cells to Xk, eliminating all relative homo* *topy groups ssi(Y; Xk) ~=ssi(Z; Xk) for small i, until we get a new finite G-complex X0 Xk* * and a map ff0: X0 -! Z extending ffk, such that Hi(Y; X0) = 0 for all i dim(X0) dim* *(Y ). Set d = dim(X0) + 1. By Lemma A.11 below, Hd(Y; X0) ~= Hd(Z; X0) is a projective Z[G]-module; and there exists a finite G-complex T such that T H = pt for all H G not of prime * *power order, and such that for some m, T is (m - 1)-connected and He*(T ) = Hm (T ) * *~= Hd(Y; X0) as Z[G]-modules. Upon replacing T by an appropriate suspension, we c* *an assume that m d, and that m - d is even. Set X00= X0_ T (recall (X0)G = XG 6= * *;), and extend ff0 to ff00: X00-! Z by sending T to a point. Then H*(Y; X00) ~=H*(Z* *; X00) vanishes except in dimensions d and m + 1, and Hd(Y; X00) ~= Hm+1 (Y; X00) as Z* *[G]- modules. Since Y and Z are connected, we can now attach (finitely many)_free or* *bits of cells Gx Di to X00, for d + 1 i m + 1, to obtain a G-map _ff: X -! Z which is_a (nonequivariant) homotopy equivalence. By construction, all isotropy subgr* *oups of X r X have prime power order. Countable case: The main difference in the proof when X and Y are countable and finite dimensional is that since we are working with countably generated mo* *dules, the group Hd(Y; X0) is stably free by Lemma A.11. So the last part of Step 2, a* *nd in particular the replacement of X0 by a wedge product, are not needed. Note that the condition XG 6= ; in Proposition 2.3 is needed only in the la* *st step of the construction, when removing the projective obstruction to making X0 -! Y* * a homotopy equivalence. If XG is empty, the calculations of projective obstruct* *ions in [OP, x4] provide other conditions under which the lemma still holds. 15 The main technical theorem on the construction of G-vector bundles can now be shown. Theorem 2.4. Assume that G is a finite group not of prime power order. Let ' : * *X -! Y be a G-map, where X is a finite G-complex and Y is countable and finite dimen* *sional. Fix a G-bundle j#X, and a P -vector bundle P #Y for each subgroup P G of prime power order, such that the following conditions hold: (1) For each prime p, the P (for p-subgroups P G) are p-locally "consistent up* * to isomorphism" with respect to morphisms in Op(G); i.e., they define an elemen* *t in the inverse limit lim- KOP (Y )(p)= lim- KOG (G=P xY )(p): G=P2Op(G) G=P2Op(G) (2) If 1 6= P G is a p-subgroup, then ['*(P )] = [j|P ] in KOP (X)(p). (3) [1] 2 KO(Y )G , and '*(1) ~=j (as nonequivariant vector bundles over X). __ Then there is_a_countable finite dimensional G-complex_X_ X such that all isot* *ropy subgroups of X rX have prime power order, a G-map__': X -! Y which extends ' and is a homotopy equivalence, a G-vector bundle _j#X, and a (real) G-representatio* *n_V , so that__j|X ~=j (V xX) as G-vector_bundles; and such that [_j] = [_'*(P ) (V xX* * )] in KO(X ) (if P = 1) or in KOP (X )(p)(if P G is a p-subgroup). If, furthermore, (4) X and Y are finite G-complexes with XG 6= ; and Y connected, (5) O(XH ) = O(Y H) for all H G not of prime power order, and fi (6) O ('-1 YiP)H = O (YiP)H for each prime pfi|G|, each p-subgroup 1 6= P G,* * each connected component YiP of Y P, and each cyclic subgroup 1 6= H=P N(P )=P of order prime to p, __ then X can be chosen to be a finite G-complex. Proof. Let f : X -! BG O be the classifying map for j (Lemma A.3). Write Y = [1* *i=1Yi, where Y1 Y2 . . .Y are finite G-subcomplexes. By Proposition 1.3 (and Lemma A.3 again), for each i, the P combine to define a G-map f0i: Yi -! B*GO which * *is unique up to G-homotopy. In particular, f0i|Yi-1 ' f0i-1for all i, and hence t* *he f0i combine to give a map fY : Y -! B*GO. Since X is a finite G-complex, conditions* * (2) and (3) (and Proposition 1.3 again) show that fY O' ' LG Of. These maps satisfy* * all of the appropriate hypotheses of Proposition 2.3. Note in particular that conditio* *n (2) in Proposition 2.3 is satisfied since the P are actual bundles (see Lemma 2.1(b)). Proposition_2.3 now applies_to give a_countable_finite dimensional (or finit* *e) G- complex X X, and G-maps _': X -! Y and f : X -!_BG O extending ' and f, such that _'is a homotopy_equivalence, such that LG Of ' fY O_', and such that_all i* *sotropy subgroups of X r X have prime power order. It remains to check_that f is induc* *ed by a G-bundle. For all H G not of prime power order, Im (ss0(f H)) = Im (ss0(* *fH )) is finite since f is induced by an actual bundle. If P G has prime power ord* *er, _ __ _f then Im(ss0(f P)) is finite since the composite X -! BG O -! BP O(p)is P -homot* *opic to 16 _ the classifying map for the P -bundle (' )*P (and ss0((BG O)P_) ~=IRO (P ) inje* *cts into ss0((BP O(p))P ) by Proposition A.2). Thus, by Lemma A.3, f factors_through BG * *O(n) for some n, and hence is the classifying map of some G-bundle _j#X, whose restr* *iction_to X is stably isomorphicftoij, and which is P -equivariantly stably isomorphic to* * '*(P ) for each prime pfi|G| and each p-subgroup P G. Theorem 2.4 can easily be combined with Theorem A.12 below, to allow the con* *struc- tion of smooth G-manifolds with various properties. But since it seems quite di* *fficult to formulate such a theorem in the greatest possible generality, we limit the appl* *ications to the case of actions on disks and euclidean spaces described in the next sect* *ion. 3. Smooth actions on disks and euclidean spaces We are now ready to describe the fixed point sets, and the tangent bundles o* *ver fixed point sets, for actions of a finite group not of prime power order on a disk or* * euclidean space. We first recall the definition of the number nG which determines which h* *omotopy types can occur among fixed point sets of actions of G on disks. Consider the set {O(XG )- 1 | X a finite contractible G-complex} Z. This* * is a subgroup of Z (as seen by taking wedge products and suspensions of G-complexes), and hence has the form nG .Z for some unique nG 0. Thus, by definition, for * *any k 2 Z, there is a finite contractible G-complex X such that O(XG ) = k if and o* *nly if k 1 (mod nG ); and the main theorem in [O1] says that any finite complex F wi* *th an "allowable" Euler characteristic can be realized as a fixed point set in this w* *ay. This is also a special case of Theorem 2.4 above: if Y is a finite contractible G-comp* *lex, and if O(F ) =_O(Y_G),_then_that theorem describes how to construct a finite contra* *ctible G-complex X with X G = F (while taking all maps to BG O and B*GO to be trivial* *). Proof of Theorem 0.1. By assumption, F is a smooth manifold, and j#F is a G- bundle such that (1) j is nonequivariantly a product bundle, (2) [j|P ] 2 gKO P* *(F ) is infinitely p-divisible for all primes p and all p-subgroups P G, and (3) jG ~* *=o(F ). Let V be the fiber over any point of F (regarded as a G-representation). Finite case: Assume that F is compact and O(F ) 1 (mod nG ). If F = ;, then nG = 1, and G has a fixed point free action on a disk by [O1, corollary to Theo* *rem 3]. So we can assume F 6= ;. By the above definition of nG , there is a finite cont* *ractible G-complex Y with O(Y G) = O(F ) (and Y G 6= ;). Set X = F _ (Y=Y G), let ' : X * *-! Y be any G-map, extend j to a G-bundle j#X by letting it be trivial over Y=Y G, a* *nd set P = (V |P )x Y for each P . Then O(XH ) = O(Y H) for all H G, and Y P is acyc* *lic and hence connected for each P G_of_prime power order. _By_Theorem 2.4, there is a finite contractible complex X X and a G-bundle _j#X_such that _j|X is st* *ably isomorphic to j. In particular, by condition (1) above, (j|F )G is stably isomo* *rphic to o(F ); and so by Theorem A.12 there is a smooth action of G on a compact contra* *ctible manifold M with fixed point set F and with o(M)|F stably isomorphic to j. By t* *he h-cobordism theorem (cf. [Mi]), M is a disk if its boundary is simply connecte* *d. So if M is not itself a disk, then we can replace it by Mx D(V ) for any G-represe* *ntation V 6= 0 with V G = 0. Countable case: Let f : F -! BG O be the classifying map for [j] - [F xV ]. * * By 17 Proposition 1.3 (and the assumptions on j), (LG Of)|F 0' * for any finite subco* *mplex F 0 F . In particular, the image of LG Of is contained in the identity connecte* *d compo- nent of (B*GO)G . By Corollary 1.4, (B*GO)G is an H-space, and (Z=n) [X; (B*GO* *)G ] is finite for any finite complex X and any n > 0. Hence Lemma A.9 applies to s* *how that there is a countable finite dimensional Z=|G|-acyclic complex Y F and a * *map fY : Y -! (B*GO)G which extends LG Of. Recall that LG : BG O -! B*GO and the forgetful map BG O -! BO are both none* *quiv- ariantly homotopy equivalences: the first by Lemma 2.1(a) and the second by Pro* *po- sition A.2(b). Let #Y be any bundle which is classified by fY : Y -! B*GO '* * BO. Then |F is a (stably) product bundle, since j#F is by assumption a product bun- dle. Let 0#(Y=F ) be an inverse bundle to (Y is finite dimensional), conside* *r it as a G-bundle over Y with trivial G-action, and let : Y -! Y=F -! (BG O)G be* * its classifying map. We can now replace [fY ] by [fY ] + [LG O ] (LG is a homomorph* *ism of H-spaces by Corollary 1.4), and thus arrange that fY : Y -! B*GO be nonequivari* *antly nullhomotopic. By Proposition 2.3, applied with X =_F_and ' : X -! Y the inclusion, there_* *is a countable_finite_dimensional G-complex X F_, together with extensions _': X * *-! Y and f : X -! BG O of ' and f, such that_X G = F and _' is nonequivariantly* * a homotopy equivalence. In_particular,_X is Z=|G|-acyclic; andfsoiby Smith theo* *ry (cf. [Br, Theorem III.7.11]),_X P is Fp-acyclic for each prime pfi|G|_and each p-su* *bgroup P G. Set d = dim(X ), and consider the d-skeleton of the join X *EG. By Lemma __ A.11, Hd (X * EG)(d) is Z[G]-stably free, and so free orbits of d- and (d + 1)-* *cells can * * __ be attached to produce a countable_finite_dimensional_contractible_G-complex Z * * X . By construction, all orbits in Zr X are free. Since X *EG is contractible, the* * inclusion __ of the d-skeleton extends to a map : Z -! X *EG(d+1). __ We next show, inductively on n, that there are G-maps fn : X *EG(n)-! BG O f* *or _ __ all n 0 which extend f : X -! BG O. Since fY is_(nonequivariantly) nullhomot* *opic and LG : BG O -!_B*GO is a homotopy equivalence, f is also nullhomotopic. So we* * can construct f0 : X *EG(0)-! BG O. __ Now assume, for some n 1, that fn-1 : X *EG(n-1)-! BG O has been constructe* *d. __ The obstruction to extending fn-1 to X *EG(n)is an element n __ def n __ ffln 2 Cn EG; gKO( X ) = Hom ZG Cn(EG); gKO( X ) : __ We can regard gKO (nX ) as the set of homotopy classes of maps of pairs i __ j (Dn; Sn-1 ) --- -! map (X ; BG O) ; (constant maps) ; and under this identification addition is given by juxtaposition of disks. Thi* *s view-_ point makes it clear that ffln_is a cocycle, and hence is a coboundary since gK* *O *(X ) is uniquely |G|-divisible (X is Z=|G|-acyclic). And if ffln is the coboundary * *of ffn-1 in 18 __ Cn-1 EG; gKO(nX ) , then ffn-1 provides the "recipe" for_changing_fn-1 on (n- * *1)- simplices in EG to obtain a map which can be extended to X *EG(n). By Lemma A.3, fd+1O : Z -! BG O induces a G-bundle _j#Z, and _j|(F =ZG ) is stably isomorphic to j. Theorem A.12 can now be applied to construct a smooth * *G- action on a contractible manifold M with fixed point set F , such that o(M)|F i* *s stably isomorphic to j. If @F = ;, then we may assume that @M = ; (otherwise just repl* *ace M by its interior). If dim(M) 5, then by [St, Theorem 5.1], M is a euclidean s* *pace if it is simply connected at infinity; in particular, if there is a sequence K1 K* *2 . .o.f compact subspaces whose union is M and such that each Mr Ki is simply connected. And if M does not satisfy this property, then Mx R does; and so we can get a eu* *clidean space by replacing M by Mx V for any G-representation V 6= 0 with V G = 0. It remains to prove Theorem 0.2; and in particular to characterize which smo* *oth manifolds have G-vector bundles which satisfy the hypotheses of Theorem 0.1. In the proofs of the next three lemmas, a pair of (real or complex) G-repres* *entations (V; W ) will be called a "P-matched pair" of type n if V |P ~=W |P for all P G* * of prime power order, and dim (V G) - dim(W G) = n. In these terms, MC MC+ MR are the classes of finite groups for which there exist a P-matched pair (V; W ) of * *complex, self-conjugate, or real G-representations, respectively, of type 1. If we only * *assume that V |P ~=W |P for p-subgroups P G (for some given prime p), then (V; W ) will be* * called a p-matched pair. Lemma 3.1. The following hold for any finite group G not of prime power order. (a) G 2 MC if and only if G contains an element not of prime power order; and G 2 MC+ if and only if G contains an element not of prime power order which is conjugate to its inverse. (b) G 2 MR if and only if there are subgroups K C H G such that H=K is dihedral of order 2n for some n not a prime power. Proof. (a) Assume g 2 G has order n = km, where k; m > 1 and (k; m) = 1. For a* *ny n-th root of unity , we let C be the 1-dimensional -representation where g* * acts via multiplication by . Set ff = exp(2ssi=k) and fi = exp(2ssi=m), and define G V = IndGC1 Cfffiand W = IndCff Cfi: Then (V; W ) is a P-matched pair of type 1, and so G 2 MC. If g is conjugate to* * g-1 , then V and W are self-conjugate _ the character of any G-representation induced* * from is real valued by the formula for the character of an induced representatio* *n (cf. [Se, x7.2, Proposition 20]) _ and so G 2 MC+ . If G 2 MC, let (V; W ) be a P-matched pair of complex representations of typ* *e 1. Then V 6~=W , but the characters of V and W agree on elements of prime power or* *der. So G must contain an element not of prime power order. Now assume that G 2 MC+ . Recall that a subgroup H G is p-elementary (for a prime p) if it is a product of a p-group with a cyclic group; and is 2-R-elem* *entary if it is a semidirect product H = Cn o P , where Cn is cyclic of order n, P is * *a 2- group, and each element of P centralizes Cn or acts on it via (g 7! g-1 ). For* * each 19 H G, let IP (H)+ R (H) be the subgroup of self-conjugate elements which vanish upon restriction to prime power order subgroups of H; or equivalently the subgr* *oup of elements whose characters are real valued and vanish on elements of prime power* * order. Then IP (H)+ is a module over RO (G), where multiplication by [V ] is induced b* *y tensor product with C RV (or by multiplication with its character). Also, the IP (H* *)+ are preserved under induction and restriction maps (by the formula for the characte* *r of an induced representation again). Since RO (G) is generated by induction from subg* *roups which are p-elementary or 2-R-elementary [Se, x12.6, Theorem 27], it now follow* *s by Frobenius reciprocity that IP (G)+ is also generated by induction from such sub* *groups (cf. [Lam, Theorem 3.4(III)]). Hence, since induction leaves unchanged the dime* *nsions of fixed point sets, there is some p-elementary or 2-R-elementary subgroup G0 G and a P-matched pair (V; W ) of self-conjugate G0-representations of type a wit* *h 2-a. In particular, G0 contains elements not of prime power order, and so G0 2 MC. T* *hus, there exist P-matched pairs of self-conjugate G0-representations of type 2; hen* *ce of type a for any a 2 Z; and so G02 MC+ . Now note that Syl2(G0) 6C G0: since otherwise 0 Syl(G0) Syl(G0) G0 dim (V G ) dim(V 2 ) = dim(W 2 ) dim(W ) (mod 2): (The representations V Syl2(G0)=V_G0_and W Syl2(G0)=W G0 of the odd order* * group G0= Syl2(G0) both split as sums U U , and hence are even dimensional, by Propo- sition A.1(c)). Hence G0 is 2-R-elementary but not 2-elementary. Write G0= Cn o* * P , where Cn = is cyclic of odd order n and P is a 2-group. We must show that * *G0 contains an element not of prime power order conjugate to its inverse, and g is* * such an element if n is not a prime power. If n > 1 is an odd prime power, and if |P* * | 4, then write g0 = gx for any element x 2 Z(P ) of order 2 which centralizes g; an* *d g0 has order 2n and is conjugate to its inverse. And the remaining possibilities _ |P * *| = 2 and n a prime power, or n = 1 _ both contradict the above observation that G0 conta* *ins elements not of prime power order. (b) Assume first that G is dihedral of order 2n, where n is not a prime power. * *Then G 2 MC by (a). Also, every CG-representation has the form C RV for some RG- representation V (this holds for any dihedral group). Thus, there is a P-matche* *d pair (C RV; C RW ) of complex G-representations of type 1, so (V; W ) is a P-match* *ed pair of real G-representations of type 1, and G 2 MR. Clearly, G 2 MR if any quotient group of G lies in MR. And if H G and (V; W* * ) is a P-matched pair of real H-representations of type 1, then (IndGH(V ); IndGH* *(W )) is a P-matched pair of real G-representations of type 1. Conversely, assume that G 2 MR. We must show that G contains a dihedral subq* *uo- tient of order 2n for some n not a prime power. The same argument as that used * *in (a) shows that G contains a 2-R-elementary subgroup which is not 2-elementary and w* *hich lies in MR. Upon replacing G by this subgroup, we may assume that G ~= Cn o P , where n is odd and P is a 2-group, where P0 = P \ CG (Cn) has index 2 in P , an* *d where elements in P rP0 act on Cn via (a 7! a-1 ). Then G=P0 is dihedral of order 2n,* * and we are done if n is not a prime power. If P is not cyclic, then let Fr(P ) denote * *its Frattini subgroup (generated by squares and commutators in P ); P= Fr(P ) is elementary * *abelian of order at least 4 (cf. [Go, Corollary 5.1.2 & Theorem 5.1.3]), and so G= Fr(P* * ) contains a subgroup which is dihedral of order 4n. 20 It remains to consider the case where P is cyclic; i.e., where fi pk 2m -1 -1 ff G = a; b fia = 1 = b ; bab = a : We must show that G =2MR. Assume otherwise: let (V; W ) be a P-matched pair of RG-representations of type 1. Decompose V and W as sums V = V11 V1x Vx1 Vxx and W = W11 W1x Wx1 Wxx; where V11 V1x = V and V11 Vx1 = V (and similarly for W ). Thus, for ex* *ample, Vxx and Wxx are the sums of those irreducible components where neither a nor b2* * acts trivially. Then 4| dim(Vxx) and 4| dim(Wxx): each irreducible real G-representa* *tion on which neither a nor b2 acts trivially is 4-dimensional (and of complex or quate* *rnion type). Since dim(V ) = dim(W ) and dim(V ) = dim(W ) by assumption, this * *shows that dim(Vx1) dim(Wx1) (mod 4). And since dim(Vx1) = 1_2dim(Vx1) (b acts o* *n each irreducible representation in Vx1 with equally many eigenvalues +1 and -1), we * *now see that dim (V G) = dim(V ) - 1_2dim(Vx1) dim(W ) - 1_2dim(Wx1) = dim(W G) (mod* * 2): And this contradicts the assumption that dim(V G) - dim(W G) = 1. The condition in Lemma 3.1(b) for G to lie in MR was pointed out to me by Er* *kki Laitinen. Recall that Fix(G) denotes the class of smooth manifolds F for which there i* *s a G- vector bundle j such that j isfnonequivariantlyia product, [j|P ] 2 gKOP (F ) i* *s infinitely p-divisible for all primes pfi|G| and all p-subgroups P G, and jG ~=o(F ). W* *e are now ready to start proving necessary and sufficient conditions for a manifold F* * to lie in Fix(G), for a given group G. The standard induction and forgetful maps between the groups of real, comple* *x, and quaternion vector bundles over F are denoted here as follows: 0 gKO(F ) ________wcu________eK(F])K________u________cSp(F * *): r q w As usual, F is called stably complex if [o(F )] 2 r(Ke(F )); or equivalently if* * o(F ) Rk has a complex structure for some k. Note that this requires that the dimensions* * of the connected components of F all have the same parity. Recall that for any abelian group A, qdiv(A) denotes the intersection of th* *e kernels of all homomorphisms from A to free abelian groups. In particular, qdiv(A) = to* *rs(A) if A is finitely generated. Lemma 3.2. Fix a finite group G not of prime power order, and a smooth manifold F . Then the following hold. (a) F 2 Fix(G) if G 2 MR, or if G 2 MC and F is stably complex. (b) F 2 Fix(G) if G 2 MC+ and c([o(F )]) 2 c0(]KSp (F )) + qdiv(Ke(F )). 21 (c) F 2 Fix(G) if [o(F )] 2 r qdiv(Ke(F )) , or if Syl2(G) 6C G and o* *(F ) 2 qdiv(KgO (F )). Proof. For any F 2 Fix(G), we let TgG (F ) denote the class of G-bundles over F* * satis- fying conditions (1)-(3) in Theorem 0.1. (a) Assume first that G 2 MR, and let (V; W ) be a P-matched pair of real G- representations such that dim(V G) = 1 and W G = 0. If F is any compact manifol* *d, let o(F ) and (F ) denote the tangent and normal bundles, and set j = o(F ) V (F ) W (* *1) (as a real G-bundle over F ). Then j|P is a product P -vector bundle for any P * * G of prime power order, and jG ~= o(F ). Thus j 2 TgG (F ), and so F 2 Fix(G). If G * *2 MC and F is stably complex, then the same construction as in (1), but with complex* * bundles and representations, again produces a bundle j 2 TgG(F ). (b) Assume G 2 MC+ and c([o(F )]) 2 c0(]KSp (F )) + qdiv(Ke(F )). By (a), the* *re are elements g; x 2 G such that |g| is not a prime power and xgx-1 = g-1 . Set G0= * *. We can choose a subgroup K C G0such that G0=K is either dihedral of order 2n wh* *ere n is not a prime power, or quaternion of order 4p for some odd prime p. In the* * first case, G 2 MR by Lemma 3.1(b), and so F 2 Fix(G) by (a). So we are reduced to the case where G0=K is quaternion of order 4p. Fix a 2 G0 which generates the cyclic subgroup of order 2p in G0=K, and set * *H = C G0. Set i = exp (2ssi=p). Then there are RG-representations V 0; W 0* *and HG-representations V 00; W 00such that C RV 0~=IndGH(C1); C RW 0~=IndGH(Ci); V 00|C ~=IndGH(C-i ); W 00|C ~=IndGH(C-1 ): Here C denotes the 1-dimensional H=K-representation where a acts via multiplic* *ation by . Set V = (C RV 0) (V 00|C) and W = (C RW 0) (W 00|C): Then (V; W ) is a self-conjugate P-matched pair of G-representations of type 1;* * and (V 0; W 0) and (V 00; W 00) are 2-matched pairs of G-representations of types 1* * and 0, respectively. Set o = o(F ), and let be a normal bundle for F . By assumption on F , ther* *e are H-bundles o00and 00such that o00 00is a product H-bundle, and such that [o00|C]* * = [C Ro] 2 eK(F )=(qdiv). Since all infinitely 2-divisible elements in eK(F )+ (* *the elements invariant under complex conjugation) are in the image of infinitely 2-divisible* * elements in ]KSp(F ) (see Lemma A.5(a)), we can assumefthatithe difference [o00|C] - [C * * Ro] is infinitely p-divisible for all odd primes pfi|G| (Lemma A.5(b)). Set j = (o RV 0) (o00 HV 00) ( RW 0) (00 HW 00): By construction, [j|P ] = 0 2 gKOP (F ) for any 2-subgroup P G. Also, C Rj = ((C Ro) C(C RV 0)) (o00 CV 00) ((C R) C(C RW 0)) (00 CW 00) (C Ro) CV (C R) CW 22 fi modulo elements which are infinitely p-divisible for all odd primes pfi|G|. So* * for each such p and each p-subgroup P G, 2.[j|P ] = r([C Rj|P ]) is infinitely p-divi* *sible, and hence [j|P ] is infinitely p-divisible by Lemma A.5(a). Thus, j 2 TgG (F )* *, and so F 2 Fix(G). (c) Assume first that G has the property that for each prime p, there is a p-ma* *tched pair (Vp; Wp) of real G-representations of type 1. We can assume that dim((Vp)G* * ) = 1 and (Wp)G = 0. LetPF be such that [o(F )] 2 qdiv(KgO (F )). By Lemma A.5(b), we* * can fi write [o(F )] = p||G|[op], where each [op] is infinitely q-divisible for all * *primes qfi|G| different from p. Choose p such that each op p is a product bundle. Set M i j j = (op Vp) (p Wp) : p||G| Then j 2 TgG(F ), and so F 2 Fix(G). If o(F ) 2 r qdiv(Ke(F )) , and if there is for each prime p a p-matched pai* *r of complex representations of type 1, then the same construction (taken with complex bundl* *es) gives a G-bundle j 2 TgG(F ). It remains to show that for each prime p, G has a p-matched pair (Vp; Wp) of* * complex representations of type 1, and a p-matched pair of real representations of type* * 1 if Syl2(G) 6C G. The complex representations are easily constructed: let g 2 Gr 1 * *be any element of order m prime to p, let C1 and Ci be the 1-dimensional -represent* *ations where g acts via multiplication by 1 or i = exp(2ssi=m), and set Vp = IndG(C1) and Wp = IndG(Ci): (* *2) fi G * * G Also, if 2fi|G| and g is any element of order 2, then Vp = Ind(R+ ) and Wp =* * Ind(R- ) form for any odd prime p a p-matched pair of real representations of type 1. If G is dihedral of order 2m, where m > 1 is odd, and if g generates the sub* *group of index 2, then the representations in (2) are induced from a 2-matched pair o* *f real G-representations. So to finish the proof, we need only show that any group G s* *uch that Syl2(G) 6C G contains a subquotient of that form. Upon dividing out by the inte* *rsection O2(G) of the Sylow 2-subgroups, we can assume that this intersection is trivial* *. Let S be any conjugacy class of elements of order 2 in G. By [Go, Theorem 3.8.2], eit* *her S generates a normal 2-subgroup of G (which is clearly not the case), or some pai* *r x; y of elements in S generates a subgroup not of 2-power order. And then is dih* *edral, and contains a dihedral subgroup of order 2m for some odd m > 1. Lemma 3.2 gave sufficient conditions for a manifold to be contained in Fix(G* *). It remains to show that these conditions are also necessary. Lemma 3.3. Fix a finite group G not of prime power order, and assume that F 2 Fix(G). (a) If Syl2(G) C G, then F is stably complex. (b) If G 62 MC, then o(F ) 2 qdiv(KgO (F )), and o(F ) 2 r qdiv(Ke(F )) if * *Syl2(G) C G. 23 (c) If G 62 MR, then c([o(F )]) 2 c0(]KSp (F )) + qdiv(Ke(F )). If G 62 MC+* * , then [o(F )] 2 r(Ke(F ))+ qdiv(KgO (F )). Proof. Fix some F 2 Fix(G). Let j#F be a G-vector bundle which satisfies condit* *ions (1)-(3) in Theorem 0.1: jG ~=fo(Fi), [j] = 0 in gKO (F ), and j|P is infinitely* * p-divisible in gKO P(F ) for each prime pfi|G| and each p-subgroup P G. (a,b) If all elements in G have prime power order, then R (G) is detected by * *re- striction to the Sylow subgroups, and so c([j]) = [C Rj] lies in qdiv(KeG(F * *)) ~= qdiv(Ke(F )) R (G). Since rOc([j]) = 2.[j], it now follows that [j] 2 qdiv(Kg* *O G(F )), and in particular that [jG ] = [o(F )] lies in qdiv(KgO (F )). Now assume that Syl2(G) C G, and write G2 = Syl2(G) for short. Then [jG2 ] is infinitely 2-divisible in gKO (F ), and hence is the image of the infinitely* * 2-divisible element c 1_2.[jG2 ] 2 eK(F ). In particular, [jG2 ] 2 r qdiv(Ke(F )) . Let V0* * = R; V1; : :V:k be the distinct irreducible real representations of G=G2. Write jG2 = j0 j1 . . .jk; where each fiber in ji is a sum of copies of Vi (in particular, j0 = jG ). Sinc* *e |G=G2| is odd, each representation V1; : :;:Vk can be given a complex structure by Pro* *position A.1(c), and hence (by Proposition A.1(a)) each ji has the form ji ~=i CVi for * *some complex bundle i#F . Thus [ji] is a complex bundle for each i 1, and so o(F ) * *~=j0 is a stably complex bundle. If, in addition, all elements of G have prime power* * order, then we have seen that [j] 2 qdiv(KgO G(F )), so [i] 2 qdiv(Ke(F )) for all 1 * *i k, and [ji] 2 r qdiv(Ke(F )) for each i 1. Also, [jG2 ] 2 r qdiv(Ke(F )) as see* *n above; and hence [o(F )] = [j0] lies in r qdiv(Ke(F )) . (c) Note first that G 2 MR (G 2 MC+ ) if there is a P-matched pair (V; W ) of* * real (self conjugate complex) representations of type a for any odd a. To see this, * *note first that if there is such a P-matched pair, then G has an element not of prime powe* *r order, and hence G 2 MC by Lemma 3.1(a). This implies that there is a P-matched pair of real G-representations of type 2; and hence a P-matched pair of real (or self c* *onjugate) G-representations of type 1. * * _ Let T : eK(F ) -! Z be any conjugation invariant homomorphism (i.e., T () = * *T ( )); and let TG denote the induced homomorphism TG = T Id: eKG(F ) ~=Ke(F ) R (G) --- -! R (G): We first analyze TG (C Rj). Let V0; V1; : :;:Vn be the distinct irre* *ducible RG- representations, where V0 ~=R is the trivial representation, and where Vi has r* *eal type for 0 i k, has complex type (with given complex structure) for k + 1 i m, a* *nd has quaternion type (with given structure) for m + 1 i n. Then iM k j i Mm j i Mn j j ~= i RVi i CVi i HVi ; i=0 i=k+1 i=m+1 24 (Proposition A.1(a)), where the i are real, complex, or quaternion vector bundl* *es, respectively, and where 0 ~= o(F ). For convenience, we write T (i) = T ([C R* *i]) if i k, T (i) = T ([i]) if k+ 1 i m, and T (i) = T ([i|C]) if m + 1 i n. Then Xk mX __ TG ([C Rj]) = T (i).c([Vi]) + (iT). [Vi] + [V i] i=0 i=k+1 Xn (* *1) + T (i).c0([Vi]) 2 R(G): i=m+1 Since (for given T ) the TG commute with restriction of subgroups, w* *e see that TG ([C Rj])|P = 0 for any P G of prime power order. Thus TG ([C Rj]) = [V ] * *- [W ], where (V; W ) is a P-matched pair of self-conjugate G-representations of type T* * (0). Assume that [o(F )] =_[0] =2r(Ke(F ))+ qdiv(KgO (F )). Then c([0]) is not a * *multiple of 2 in Ke(F )=<[] | [ ] 2 (qdiv)>. So by Lemma A.5(c), we can choose T such* * that T (0) is odd. Then TG ([C Rj]) = [V ] - [W ] where (V; W ) is a P-matched pai* *r of self-conjugate G-representations of type T (0), and G 2 MC+ by the above remark* *s. If c([o(F )]) = c([0]) =2c0(]KSp (F )) + qdiv(Ke(F )), then qOc([0]) is not * *a multiple of 2 in ]KSp(F )=(qdiv). By Lemma A.5(c) again, we can choose T to be a composite * *of the form eK(F ) -q!]KSp(F ) -! Z, and such that T (0) is odd. In particular, T * *(i) 2 2Z for m + 1 i n (the quaternion case). By (1), TG ([C Rj]) = c([V ] - [W ]), w* *here Xk mX Xn T ( ) [V ] - [W ] = T (i).[Vi] + T (i).r([Vi]) + ___i_.rOc0([Vi]* *); i=0 i=k+1 i=m+1 2 and (V; W ) is a P-matched pair of real G-representations of type T (0). It fol* *lows that G 2 MR. We now get immediately: Proof of Theorem 0.2. If F = MG for any contractible manifold M with smooth G-action, then the G-vector bundle j = o(M)|F satisfies conditions (1)-(3) in T* *heorem 0.1, and so F 2 Fix(G). Also, O(F ) 1 (mod nG ) if M is a disk. Conversely* *, by Theorem 0.1, if F 2 Fix(G), then F is the fixed point set of a smooth G-action * *on a euclidean space if @F = ;, and F is the fixed point set of a smooth G-action on* * a disk if F is compact and O(F ) 1 (mod nG ). The necessary and sufficient conditions for F to be in Fix(G) were shown in * *Lemmas 3.2 and 3.3. Note in particular case (C) in Theorem 0.2. If G 2 MCr MC+ and Syl2(G) 6C G, then F 2 Fix(G) if [o(F )] 2 r(Ke(F )) (Lemma 3.2(a)) or if [o(F* * )] 2 qdiv(KgO (F )) (Lemma 3.2(c)). So from the definition of Fix(G), it follows th* *at F 2 Fix(G) if [o(F )] 2 r(Ke(F )) + qdiv(KgO (F )). The following example shows how Theorem 0.2 applies in the case of a dihedra* *l or quaternion group acting on a disk. 25 Example 3.4. If G is dihedral of order 2n or quaternion of order 4n, where n is* * not a prime power, then a compact manifold F is the fixed point set of a G-action on * *some disk if and only if O(F ) is odd. If G is dihedral of order 2pa for some odd pr* *ime p, then a compact manifold is the fixed point set of a G action on a disk if and only i* *f O(F ) = 1 and [o(F )] is torsion in gKO (F ). If G is quaternion of order 4pa for some o* *dd prime p, then a compact manifold is the fixed point set of a G action on a disk if an* *d only if O(F ) = 1, and there is an H-vector bundle #F such that c([o(F )]) c0([]) m* *odulo torsion in eK(F ). Proof. If G is dihedral of order 2n or quaternion of order 4n, then by Theorem * *0.3, nG = 2 if n is not a prime power, and nG = 0 if n is a power of an odd prime. T* *he rest of the corollary follows from Lemma 3.1 and Theorem 0.2. As another example, note that for G = A4x 3, any compact smooth manifold F c* *an be the fixed point set of a smooth G-action on a disk. This is in fact the smal* *lest group with that property (see [O1, Theorem 8]). Appendix We collect here some results which are well known, but which either are hard* * to find in the literature, or which have been used often enough to state here explicitl* *y. Real G-vector bundles and their classifying spaces We start with the following proposition, which describes some of the basic s* *tructure of real G-vector bundles and real G-representations. Proposition A.1. Fix a finite group G. Let V0; V1; : :;:Vk be the distinct irre* *ducible RG-representations, where V0 ~= R with the trivial G-action. For each i, set D* *i = End RG (Vi) (~=R, C, or H). (a) LetLX be space with trivial G-action, and let #X be a real G-vector bund* *le. Then ~= ki=0(Vi Dii), where each i is a (nonequivariant) Di-vector bundle o* *ver X. (b) Let V be any orthogonal G-representation, and let OG (V ) be the group * *of G- equivariant orthogonal self maps of V . Then Mk Yk V ~= (Vi)ni implies OG (V ) ~= O(ni; Di); i=0 i=0 where we write O(n; R) = O(n), O(n; C) = U(n), and O(n; H) = Sp(n). (c) If |G| is odd, then Di ~=C for all i 6= 0. Proof. For each i, End RG(Vi) is a division algebra over R by Schur's lemma (cf* *. [Ad, Lemma 3.22]), and hence is isomorphic to R, C, or H. Part (b) also follows from* * Schur's lemma, and part (c) from [Se, Exercise 13.12]. 26 To see part (a), set i = Hom RG (Vi; ) (defined fiberwise) for each i. Then* * i is a Di-bundle, and the evaluation maps define an isomorphism Mk ~ Vi DiHom RG (Vi; ) --=--! : i=0 An irreducible RG-representation V will be said to have real, complex, or qu* *aternion type, depending on whether End RG(V ) is isomorphic to R, C, or H. For each n 0, BG O(n) will denote the classifying space for n-dimensional G* *-vector bundles: constructed using either infinite joins (cf. [tD, xI.8]), or Grassma* *nnians of n-dimensional subspaces in an appropriate infinite dimensional G-representation* *. It has a universal G-vector bundle EnG#BG O(n) with respect to which pullback defi* *nes a bijection between [X; BG O(n)]G and the set of locally trivial n-dimensional or* *thogonal G-bundles over X, for any countable G-complex X (cf. [tD, Theorem I.8.12], whe* *re the classifying space is denoted B(G; O(n))). Note that BG O(n) is connected fo* *r all n, since ss0(BG O(n)) contains just one element: the class of the product bundle G* *x Rn#G. For each orthogonal G-representation V , and each m 0, direct sum with V de* *fines a G-map V : BG O(m) ----! BG O(m+ dim (V )); which is well defined up to G-homotopy. We define BG O to be the homotopy dire* *ct limit (i.e., infinite mapping cylinder, or mapping telescope) i j BG O = hocolim-----!BG O(0) --RG-!BG O(d) --RG-!BG O(2d) --RG-!: :;: where RG denotes the regular representation and d = dim(RG) = |G|. For each n, * *we let n : BG O(nd) -! BG O denote the inclusion of the n-th stage into this telescope. If X is any finite G-complex, then any map X -! BG O factors through some fi* *nite stage BG O(nd) in the mapping telescope, and similarly for homotopies between m* *aps. Hence i * * j [X;BG O] ~=lim-![X; BG O(0)] --RG-![X; BG O(d)] --RG-![X; BG O(2d)] --RG-!:* * : : i j ~=lim VectR;G(X) --RG-!VectR;G(X) --RG-!VectR;G(X) --RG-!: : : -! 0 d 2d (d = |G* *|) ~=Ker KOG (X) -dim-!Z ; Here, VectR;Gm(X) denotes the set of isomorphism classes of m-dimensional ortho* *gonal G-vector bundles over X; and the last step holds since any G-vector bundle over* * X is a summand of a product bundle RGkx X for some k (since any G-representation * *is contained in some multiple of the regular representation RG). In particular, th* *is shows that Zx BG O is the classifying space for the equivariant K-theory functor KOG * *(-). We next look more closely at the fixed point sets (BG O)H and (BG O(n))H . 27 Proposition A.2. Fix a finite group G and a subgroup H G. (a) For each n 0, a (BG O(n))H ' BOH (V ) [V ]2RepRn(H) where Rep Rn(H) is the set of isomorphism classes of n-dimensional orthogonal H- representations. (b) Let V0; V1; : :;:Vk be the distinct irreducible orthogonal H-representat* *ions. Then Yk (BG O)H ' IRO (H) x Bi; i=0 where IRO (H) = Ker[RO (H) -dim-!Z] is the augmentation ideal, and where Bi ~=B* *O, BU, or BSp depending on whether End RG(Vi) ~=R, C, or H. H (c) For any n > 0, Hn : BG O(nd) -! (BG O)H (d = |G|) sends the compone* *nt H BG O(nd) V corresponding to the representation V to the component of (BG O)H * *cor- responding to [V ] - [RGn] 2 IRO (H); and the map between this pair of componen* *ts is m-connected if each irreducible H-representation occurs in V with multiplicity * *at least m. (d) For any finite H-complex X, ( -i KOH (X) if i > 0 ssi map H (X; BG O) ~= dim Ker KOH (X) --! Z if i = 0. Proof. Consider the H-equivariant maps BG O(n) -f1!BH O(n) -f2!BG O(n), where f1 classifies the universal bundle EnG#BG O(n) regarded as an H-bundle, and where * *f2 classifies the G-bundle (Gx HEnH)#(Gx HBH O(n)). These are easily checked to be* * H- homotopy inverses; and show that BG O(n) is H-equivariantly homotopy equivalent* * to BH O(n). So BG O is H-homotopy equivalent to BH O. In particular, it suffices t* *o prove the proposition when H = G. (a) For any n, ss0((BG O(n))G ) ~=VectR;Gn(pt) ~=Rep Rn(G): Let BG O(n) GVdenote the component corresponding to the representation V . For* * any X (without group action), [X; (BG O(n))G ] ~=[X; BG O(n)]G ~=Vect R;Gn(X); and so [X; (BG O(n))GV] is the set of isomorphism classes of G-vector bundles o* *ver X with fiber V . The structure group for such bundles is OG (V ), and hence (BG O* *(n))GV' BOG (V ). 28 (b,c) The descriptions of the components of (BG O)G , and of Gn : (BG O(nd))* *G -! (BG O)G , follow from part (a) and Proposition A.1(b), upon taking limits with * *respect to direct sum with the regular representation RG. Note in particular that i j IRO(G) ~=lim-!RepR0(G) --RG-!RepRd(G) --RG-!RepR2d(G) --RG-!: ::: And the last statement in (c) follows since the inclusions BO(m) -! BO, BU(m) -* *! BU, and BSp(m) -! BSp are m-connected for all m. (d) We have already seen that [X; BG O]G ~=Ker [KOG (X) -dim-!Z] when X is a* * finite G-complex. And for any i > 0, i G -i ssi map G(X; BG O) ~=[ (X+ ); BG O]* ~=KOG (X): As was noted above, any map from a finite G-complex X to BG O factors through some n : BG O(nd) -! BG O, and hence induces (stably) a G-bundle over X. This d* *oes not hold in general for finite dimensional G-complexes, but the next lemma desc* *ribes conditions under which maps X -! BG O do induce G-bundles. Lemma A.3. Fix a countable finite dimensional G-complex X. Then for each n 0, pullback of the universal bundle EnG#BG O(n) defines a bijection between [X; BG* * O(n)]G and the set of isomorphism classes of n-dimensional G-bundles over X. Also, a G* *-map f : X -! BG O factors through m : BG O(md) -! BG O for some m (where d = |G|) if and only if Im (ss0(fH )) ss0((BG O)H ) is finite for all H G. And any two* * liftings fm ; f0m : X -! BG O(md) of f are homotopic after some finite stabilization; i* *.e., the induced G-bundles over X are stably isomorphic. Proof. The bijection between n-dimensional G-bundles over X and [X; BG O(n)]G i* *s a special case of [tD, Theorem I.8.12]. If f : X -! BG O factors through some BG O(md), then Im(ss0(fH )) must be fi* *nite for all H since (BG O(md))H has only finitely many connected components (correspon* *ding to the finite set of m-dimensional H-representations). Conversely, set n = dim* * (X), and assume that Im (ss0(fH )) is finite for all H G. For each H, we can choose* * some mH 0 large enough so that the image of fH is contained in components of (BG O* *)H corresponding to some family of virtual H-representations vi = [Vi]-[RGmH ] 2 I* *RO (H) (1 i k), and such that each irreducible H-representation occurs in each Vi wi* *th multiplicity at least n. Thus, the image of any connected component of XH is l* *ies in one of the components (BG O)Hvi, which is in the image of (BG O(mH .d))HVi; and* * the inclusion of those components is n-connected by Proposition A.2(c). Hence, if * *we set m = max {mH | H G}, then f : X -! BG O factors through BG O(md). Divisible and quasidivisible subgroups The purpose of the following two lemmas is to set up some notation and resul* *ts to work with cohomology and K-theory groups of countably infinite CW complexes. He* *nce, we concentrate on the class of what we call PFG -groups ("pro-finitely generat* *ed"): abelian groups which are products of the form lim-(Mi) x lim-1(M0i), where Mi a* *nd M0i are two inverse systems of finitely generated abelian groups. We first note tha* *t the lim-1 factor is divisible (i.e., n-divisible for all n > 0). 29 i j Lemma A.4. Fix a sequence . . .-!M2 -! M1 -! M0 of abelian groups. Then for any n > 0, lim-1(Mi) is n-divisible if Mi=nMi is finite for all i. In parti* *cular, if the Mi are all finitely generated, then lim-1(Mi) is divisible, and hence injective. Proof. Since lim-1is right exact, the sequence lim-1(Mi) --.n--!lim-1(Mi) ----! lim-1(Mi=nMi) -! 0 i i i is exact for all n > 0, and lim-1(Mi=nMi) = 0 if the Mi=nMi are finite. So lim-* *1(Mi) is n-divisible in this case. For any abelian group A, we define qdiv(A) to be the smallest possible kerne* *l of a homomorphism from A to a product of copies of Z. Clearly, all elements in A wh* *ich are infinitely p-divisible for any prime p are contained in qdiv(A). So by Lemm* *a A.4, if A = lim-(Mi) x lim-1(M0i) where Mi and M0iare inverse systems of finitely ge* *nerated abelian groups, then qdiv(A) = lim-(tors(Mi)) x lim-1(M0i). Lemma A.5. If X is a countable CW complex, and if h* is any (representable) coh* *o- mology theory such that hi(pt) is finitely generated for all i, then hi(X) is a* * PFG -group for all i. In particular, eK(X), gKO(X), and ]KSp(X) are all PFG -groups. Furt* *hermore, the following hold for any PFG -group A: (a) If x 2 A is divisible, i.e., if x 2 nA for all n > 0, then x is "sequent* *ially divisible" in that for any sequence n1; n2; : :o:f positive integers there is a sequence x = * *x0; x1; x2; : : : in A such that nixi = xi-1 for all i. Similarly, if x 2 A is infinitely p-divis* *ible for any prime p, then there is a sequence x = x0; x1; x2; : :s:uch that pxi = xi-1 for * *all i. And if nx is infinitely p-divisible for any prime p-n, then x is also infinitely p-* *divisible. P (b) For any n, and any x 2 qdiv(A), we can write x = p|nxp, where each xp * *is infinitely q-divisible for all primes q 6= p dividing n. (c) If x 2 A, and x =22A+ qdiv(A), then there is a homomorphism ' : A -! Z s* *uch that '(x) is odd. Proof. Fix a countable CW complex X, and write X = [ 1i=1Xi, where X1 X2 X3 . . .are finite subcomplexes. Then for any representable cohomology theory * *h*, there is for each j a short exact sequence j-1 j j 0 -! lim-1eh (Xi) ----! h (X) ----! lim-h (Xi) -! 0; i i where the hj(Xi) and ehj-1(Xi) are all finitely generated. The extension split* *s, since the first term is injective by Lemma A.4, and so hj(X) is a PFG -group. Now assume A is a PFG -group, and write A = lim-(Mi)xlim-1(M0i), where the * *Miand M0iare all finitely generated. Point (a) follows upon noting that the divisible* * elements in A are precisely those in lim-1(M0i), and that the p-divisible elements (for * *any prime p) are those in lim-p0-tors(Mi) x lim-1(M0i). Point (b) is immediate. Point (c* *) follows upon noting that if x =22A + qdiv(A), then the image of x in some Mi=(tors) is * *not a multiple of 2. And then there is a homomorphism Mi=(tors) -! Z which sends t* *he image of x to an odd integer. 30 Homotopy and homology groups We collect here some miscellaneous lemmas on homotopy and homology groups and the Hurewicz map. Lemma A.6. All homotopy groups of a countable CW complex are countable. Proof. The homotopy groups of a finite simply connected complex are finitely ge* *nerated (cf. [Hu, Corollary X.8.3]). The fundamental group of a countable complex is co* *untably generated and hence countable. So the homotopy groups of the universal cover o* *f a countable complex are countable direct limits of finitely generated groups, and* * hence are countable. The following version of the relative Hurewicz theorem is needed in Section * *2 when constructing spaces and maps. For convenience, when a map f : X -! Y is underst* *ood, we write ss*(Y; X) for ss*(Zf; X) (where Zf denotes the mapping cylinder), and * *similarly for H*(Y; X). Lemma A.7. Fix a prime p and n 2. Assume that f : X -! Y is a map between connected complexes such that ss1(f) is onto, such that Ker(ss1(f)) is abelian * *and torsion prime to p, and such that ssi(Ye; eX) Z(p)= 0 for all i < n. Here, Xe and eY * *denote the universal covers. Then Hi(Y; X) Z(p)= 0 for all i < n, and the Hurewicz m* *ap ssn(Ye; eX) Z(p)i Hn(Y; X; Z(p)) is onto. If, furthermore, X and Y are finite * *complexes and Ker (ss1(f)) is finite, then ssn(Ye; eX) Z(p) is finitely generated as a Z* *(p)[ss1(X)]- module. (Note that ss2(Ye; eX) ~=Im [ss2(Y ) -! ss2(Y; X)], and ssi(Ye; eX) = ssi(Y; X)* * for i > 2. The lemma is formulated using ssi(Ye; eX) rather than ssi(Y; X) to allow the possib* *ility that ss2(Y; X) is not abelian.) Proof. Let F be the homotopy fiber of f : X -! Y , and let eFbe its universal c* *over (F is connected by assumption). Then ssi(Fe) Z(p)~=ssi+1(Y; X) Z(p)= 0 for all 2 i < n- 1. So by the generalized Hurewicz theorem (cf. [Hu, Theorem X.8.1]), applied to th* *e class of torsion abelian groups of order prime to p, Hi(Fe; Z(p)) = 0 for all i < n- 1and Hn-1 (Fe; Z(p)) ~=ssn-1 (Fe) Z(* *p):(1) Set h i = ss2(Ye; eX) ~=Im ss2(Y ) --! ss2(Y; X)~=ss1(F ) = Ker ss1(F ) -! ss* *1(X) ; __ and let F = Fe= be the covering of F with fundamental group . By Lemma A.8 below, acts trivially on H*(Fe). If n 3, then is abelian and_torsion prime t* *o p, by assumption, and the spectral sequence for the fibration eF-! F -! B gives __ H*(F ; Z(p)) ~=H0(; H*(Fe; Z(p))) ~=H*(Fe; Z(p)): 31 Together with (1), this shows that __ __ ssi(F ) Z(p)~=Hi(F ; Z(p)) for all i n - 1 (* *2) whenever i 2, and this clearly also holds for i = 1. __ Since Hi(F ; Z(p))_= 0 = Hi(Fe; Z(p)) for i < n-1 by (1) and (2), the spectr* *al sequence for the fibration F -! Xe -! eYshows that __ Hi(Ye; eX; Z(p)) = 0 for i < n and Hn(Ye; eX; Z(p)) ~=Hn-1 (F ; Z(* *p)):(3) And this together with (2) shows that the p-local Hurewicz homomomorphism for t* *he pair (Ye; eX) is a composite of isomorphisms: __ __ ssn(Ye; eX) Z(p)~=ssn-1 (F ) Z(p)~=Hn-1 (F ; Z(p)) ~=Hn(Ye; eX; Z(p)* *):(4) Set K = Ker(ss1(f)). By assumption, K is abelian and torsion prime to p. H* *ence H*(Ye; eX=K; Z(p)) ~=H0(K; H*(Ye; eX; Z(p))). Since Hi(Ye; eX; Z(p)) = 0 for i * *< n by (3), the spectral sequence for the fibration (Ye; eX=K) -! (Y; X) -! ss1(Y ) shows t* *hat Hi(Y; X; Z(p)) ~=H0 ss1(Y ); Hi(Ye; eX=K; Z(p)) ~=H0(ss1(X); Hi(Ye; eX; Z* *(p))) for all i n. Together with (3) and (4), this shows that Hi(Y; X; Z(p)) = 0 for* * i < n, and that the Hurewicz homomorphism sends ssn(Ye; eX) Z(p)onto Hn(Y; X; Z(p)). Now assume that X and Y are finite complexes, and that Ker[ss1(X) i ss1(Y )]* * is fi- nite. The kernel has order prime to p, by assumption, and so any projective Z(p* *)[ss1(Y )]- module is also projective as a Z(p)[ss1(X)]-module. Each term in the relative * *cellular chain complex C* = C*(Ye; eX; Z(p)) is thus a finitely generated projective Z(p* *)[ss1(X)]- module. Since C* has no homology below dimension n, Zn def=Ker[Cn -@!Cn-1 ] is* * a direct summand of Cn and hence finitely generated. So Hn(Ye; eX; Z(p)) = Zn=@(C* *n+1 ) is also finitely generated over Z(p)[ss1(X)]; and is isomorphic to ssn(Ye; eX) * * Z(p)by (4). It remains to prove the following lemma, which says in particular that the h* *omotopy fiber of a map between connected and 1-connected spaces is simple. Lemma A.8. Let F -! X -f!Y be a fibration of path connected spaces such that F has a universal cover eF; and set = Ker[ss1(F ) -! ss1(X)]. Then the translati* *on action of any element of on eFis homotopic to the identity. In particular, acts triv* *ially on ss*(Fe) and on H*(Fe). Proof. Fix a basepoint x0 2 F X, and set y0 = f(x0) 2 Y . Let fl : I -! F be any loop (fl(0) = fl(1) = x0) which represents an element of , and choose a hom* *otopy G : Ix I -! X such that G(t; 0) = fl(t), and G(t; s) = x0 if s = 1 or t 2 {0; 1* *}. Then [f O G] 2 ss2(Y ), and @([f O G]) = [fl]. 32 Define ff = f O G O proj: F xIx I -! B. By the homotopy lifting property fo* *r the fibration, there exists a map A : F xIx I -! X such that A(x; t; s) = x if s =* * 1 or t 2 {0; 1}. Let fi : F xI -! F be the map fi(x; t) = A(x; t; 0). Then fi(-; 0) * *= fi(-; 1) = IdF , and so fi can be lifted to a unique homotopy efi: eFxI -! eFsuch that efi* *(-; 0) = Id. Also, the loop fi(x0; -) is homotopic to fl by construction, so efi(-; 1) is th* *e covering transformation induced by fl, and is thus homotopic to the identity. The next lemma is much more technical. It is needed in the proof of Theorem * *0.1, to handle fixed point sets not of finite homotopy type. Lemma A.9. Fix n > 0, and let B be a connected H-space with the property that (Z=n) [K; B] is finite for any finite CW complex K. Let X be a countable fini* *te dimensional complex, and let f : X -! B be a map which is nullhomotopic on all * *finite __ _f subcomplexes of X. Then f factors as a composite_X ,! X -! B, for some countab* *le finite dimensional Z=n-acyclic complex X X. Proof. Write X = [1i=1Xi, where X1 X2 X3 . .a.re all finite subcomplexes. By assumption, [f] 2 [X; B] lies in the image of the first term in the short exact* * sequence 0 -! lim-1[(Xi); B] ----! [X; B] ----! lim-[Xi; B] -! 0: By Lemma A.4, the group lim-1[(Xi); B] is n-divisible, and so there is a sequen* *ce of maps f = f0; f1; f2; : :::X -! B such that n.[fi] = [fi-1] for each i. Hence f * *factors as a composite bf X ----! bB----! B; i .n .n .n j where bBis the homotopy inverse limit of the sequence : :-:! B -! B -! B . We next claim that Bb is Z=n-acyclic. To see this, note that each map B -.n* *!B induces multiplication by n in homotopy groups, and so all homotopy groups of b* *Bare uniquely n-divisible. Hence, via spectral sequences for the fibrations, it will* * suffice to show that He*(K(M; i); Z=n) = 0 for all i 1 and all Z[ 1_n]-modules M. It suff* *ices (by taking direct limits) to show this for finitely generated M; and hence for M = * *Z[ 1_n] and M finite of order prime to n. The latter case is clear. When M = Z[ 1_n],* * then K(M; 1) is a Z[ 1_n]-Moore space (and hence Z=n-acyclic); and its deloopings ar* *e seen to be Z=n-acyclic using the usual spectral sequences. It_remains_to show that bfextends to a countable finite dimensional Z=n-acyc* *lic com- plex X X. To see this, first replace bBby a CW complex (of the same weak homot* *opy type) which contains X as a subcomplex. Since homology is supported by finite c* *om- plexes, there is a sequence X = X0 X1 X2 . .o.f countable subcomplexes of bB such that each inclusion Xi-1 Xiis trivial in Z=n-homology. Set X1 = [1i=1Xi.* * Then X1 is countable and Z=n-acyclic, but need not be finite dimensional. Set d = d* *im(X), and consider the free abelian group Bd(X1 ) def=Ker[Hd((X1 )(d)) -! Hd(X1 )]. E* *very element in Bd(X1 ) is in the image of the_Hurewicz homomorphism for (X1 )(d). * *So there is a (d + 1)-dimensional complex X with the same d-skeleton as X1 , such* * that 33 __ __ __ the_inclusion_X (d) X1 extends_to_X , and such that Hi(X ) ~=Hi(X1_) for i d * *and Hi(X ) = 0 for i > d. Then eH*(X ) is uniquely n-divisible, so X is Z=n-acycli* *c, and we are done. Projective and stably free homology of G-complexes The following lemma is basically taken from [O1], although not stated there * *explicitly. Lemma A.10. Let G be any finite group, and let f : X -! Y be a map between countable finite dimensional G-complexes. Set n = max {dim (X); dim(Y )}. Ass* *ume, for some prime p, that Hei(Zf; X; Fp) = 0 for all i n, and that fP : XP -! Y* * P is an Fp-homology equivalence for all p-subgroups 1 6= P G. Then Hn+1 (Zf; X; Fp) is projective as an Fp[G]-module, and hence is stably free as a countably gener* *ated Fp[G]-module. If in addition, X and Y are finite complexes, and O(XH ) = O(Y H)* * for any cyclic subgroup 1 6= H G of order prime to p, then Hn+1 (Zf; X; Fp) is a f* *ree Fp[G]-module. Proof. By replacing Y with the mapping cone of f, we can assume that X is a po* *int (and dim (Y ) n+ 1). Throughout the proof, for any subcomplex Y 0 Y , we let C*(Y; Y 0; Fp), denote the cellular chain complex of (Y; Y 0): the complex who* *se n-th degree term is the free Fp-module with one generator for each n-cell in Y not i* *n Y 0. Fix a Sylow p-subgroup S G, and let Ys be the union of the fixed point sets Y P taken over all nontrivial subgroups 1 6= P S. Then Ys is a union of Fp-ac* *yclic subcomplexes such that all intersections are also Fp-acyclic. Hence Ys is itsel* *f Fp-acyclic (seen using Mayer-Vietoris sequences). We thus get an exact sequence 0 -! Hn+1 (Y ; Fp) -! Cn+1 (Y; Ys; Fp) -! Cn(Y; Ys; Fp) -! : :-:!C0(Y; Ys; F* *p) -! 0; and each term Ci(Y; Ys; Fp) is free as an Fp[S]-module since S acts freely on Y* * rYs. Thus, Hn+1 (Y ; Fp) is stably free as an Fp[S]-module; and hence is projective * *as an Fp[G]-module (cf. [Rim, Corollary 2.4 & Proposition 4.8]). And by the "Eilenb* *erg swindle", any countably generated projective module M is stably free in the cat* *egory of countably generated modules: we can write M N ~=F for some countably generated free module F , and then M F 1 ~=M (N M) (N M) . .~.=(M N) (M N) . .~.=F 1: Assume now that X and Y are finite complexes, and that O(Y H) = 1 for any cy* *clic subgroup 1 6= H G of order prime to p. We want to show that Hn+1 (Y ; Fp) is F* *p[G]- free. In general, by [Se, x16.1], two projective Fp[G]-modules M1 and M2 are is* *omorphic if and only if [M1] = [M2] 2 RFp(G) (the representation ring of all finitely ge* *nerated Fp[G]-modules modulo short exact sequences). By [Se, x18.2], this is the case * *if and only if M1 and M2 have the same modular character, where the modular character * *of an Fp[G]-module is a complex valued function defined on the set of elements of G o* *f order prime to p. Thus, a projective Fp[G]-module M is free if and only if M ~=(Fp[G]* *)s for s = rkFp(MG ), if and only if M|G|~= (Fp[G])r for r = rkFp(M), if and only if M* * is free as a Fp[H]-module for each cyclic subgroup H G of order prime to p. And for any 34 such H, since O(Y K) = 1 for all 1 6= K H by assumption, a count of the number* *s of H-orbits of cells in Y shows that in RFp(H), Xn (-1)n[Hn+1 (Y ; Fp)] = (-1)i[Ci(Y; pt; Fp)] 0 (mod ): i=0 The next lemma provides an analogous result for integral homology. Lemma A.11. Let G be any finite group, and let f : X -! Y be a map between countable finite dimensional G-complexes. Set n = max {dim (X); dim(Y )}. Ass* *ume that Hei(Zf; X; Z) = 0 for allfii n, and that fP : XP -! Y P is an Fp-homology equivalence for all primes pfi|G| and all p-subgroups 1 6= P G. Then Hn+1 (Zf;* * X; Z) is Z[G]-projective, and hence is stably free as a countably generated Z[G]-modu* *le. If in addition X and Y are finite complexes, and O(XH ) = O(Y H) for all 1 6= H G, t* *hen there is a finite G-complex T such that T H = pt for all H G not of prime power order, and such that for some d, T is (d- 1)-connected and He*(T ; Z) = Hd(T ; * *Z) ~= Hn+1 (Zf; X; Z) as Z[G]-modules. Proof. By LemmafA.10,iHn+1 (Zf; X; Fp) = Z=p Hn+1 (Zf; X) is Fp[G]-projective * *for all primes pfi|G|. Also, Hn+1 (Zf; X; Z) is Z-free, since dim (Zf) = n+ 1. In p* *articular, Hn+1 (Zf; X; Z) is G-cohomologically trivial; and hence is Z[G]-projective by R* *im's theorem [Rim, Theorem 4.11]. And by the "Eilenberg swindle" again, this implies* * that Hn+1 (Zf; X; Z) is Z[G]-stably free in the category of countably generated Z[G]* *-modules. Now assume that X and Y are finite complexes. Let Cf denote the mapping cone of f : X -! Y , so Hn+1 (Cf) ~=Hn+1 (Zf; X) is Z[G]-projective. Let CNPf be the* * set of elements x 2 Cf whose isotropy subgroup Gx does not have prime power order. Then O((CNPf)H ) = 1 for all H G (since O(XH ) = O(Y H) when H 6= 1, and all free orbits have been removed). Hence, by [O2, Proposition 5], there is a finite con* *tractible G-complex Z CNPf such that all isotropy subgroups of Zr CNPf have prime power order. By [O2, Lemma 11], applied to the inclusion Z Z[ CNPfCf, there exists a* * finite contractible G-complex Z0 Cf such that all isotropy subgroups of Z0r Cf have p* *rime power order. If we now set T = Z0=Cf, then T ' (Cf) (nonequivariantly); and so T is (n+ 1)-connected and eH*(T ) = Hn+2 (T ) ~=Hn+1 (Zf; X). Since this last argument is rather indirect, we now outline a more direct ar* *gument to help explain what is really going on. The details are similar to those used * *in [O1, x3] to study fixed point sets. For any G-complex X, XNP X denotes the union of fi* *xed point sets of subgroups not of prime power order. A finite G-complex X will be * *called simple if O(XH ) = 1 for all H G. A finite G-complex X will be called a G-reso* *lution if X is n-dimensional and (n- 1)-connected for some n, and Hn(X) is Z[G]-projec* *tive. For any G-resolution X, set flG (X) = (-1)n[Hn(X)] 2 Ke0(Z[G]) (n = dim (X)). * *Let B0(G) eK0(Z[G]) be the subset of all flG (X) for G-resolutions X such that XNP* * is a point. Using direct geometric constructions, one now shows that B0(G) is a subg* *roup, and that there is a well defined function G : {finite simple G-complexes} --- -! eK0(Z[G])=B0(G) 35 __ __ __ which sends X to flG (X ) for any G-resolution X such that X NP ~= XNP . Also, * *G (X) depends only on the G-homotopy type of X, and G is additive in the sense that G (Y=X) = G (Y ) - G (X) for any pair X Y . It is now straightforward to show that any such function from finite simple G-complexes to an abelian group is tr* *ivial. And when applied to the mapping cone Cf defined above (more precisely, to CNPf)* *, this shows that [Hn+1 (Zf; X)] 2 B0(G). An equivariant thickening theorem The procedure for constructing manifolds with smooth G-action, starting with* * a G- vector bundle over a countable finite dimensional G-complex, is based on the fo* *llowing equivariant thickening theorem. As has been seen, it provides a good tool when* * con- structing smooth G-actions on disks and euclidean spaces. In contrast, it cann* *ot be used (at least not directly) to construct smooth actions on closed manifolds. Theorem A.12. [Pawalowski] Fix a finite group G, a countable finite dimensional* * G- complex X, and a G-vector bundle #X. Assume that F = XG is given the structure of a smooth manifold, and that (|F )G is stably isomorphic to the tangent bund* *le o(F ). Then there is a smooth manifold M with smooth G-action, containing X as* * a G-deformation retract, such that MG is diffeomorphic to F , and such that o(M)* *|X is stably G-isomorphic to (i.e., (o(M)|X) (V xX) ~= (W xX) for some pair of G- representations V and W ). If X is a finite G-complex, then M can be chosen t* *o be compact. Proof. See [Pa2, Theorems 2.4 & 3.1] (where the result is stated more precisely* *). The idea of the proof is the following. After adding a product bundle to , we can a* *ssume that G #F ~=o(F ) (Rkx F ) for some k; and that dim ((x)H ) > 2.dim (XH ) + k and dim ((x)H ) - dim((x)> H) > dim(XH ) for all H $ G and all x 2 XH . Here, (x)> H denotes the union of the fixed poin* *t sets of subgroups of Gx strictly containing H. Choose G-invariant subcomplexes F = X0 X1 X2 . .X., such that X = [Ni=0Xi (where N 1), and such that each Xi is obtained from Xi-1 by attaching one orbit of cells G=H x Dj for some H and j. Manifolds M0 M1 M2 . . . are now constructed such that for each i, (Mi)G = F , Xi Mi and (@Mir @F ) (Mir Xi) are G-deformation retracts, and o(Mi)|Xi (Rkx Xi) ~= |Xi. To start t* *he procedure, let M0 be the disk bundle of (|F )=(G ). The induction step is carri* *ed out using standard (nonequivariant) embedding theorems, and theorems about destabil* *izing vector bundles and isomorphisms between them. The manifold M = [Ni=0Mi now satisfies the conclusions of the theorem. References [Ad] J. F. Adams, Lectures on Lie groups, Benjamin (1969) [Br] G. Bredon, Introduction to compact transformation groups, Academic Press (* *1972) [tD] T. tom Dieck, Transformation groups, de Gruyter (1987) 36 [EL] A. Edmonds & R. Lee, Topology 14_(1975), 339-345 [Go] D. Gorenstein, Finite groups, Harper & Row (1968) [Hu] S.-T. Hu, Homotopy theory, Academic Press (1959) [JM] S. Jackowski & J. 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